Greatest common divisor
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Find the greatest common divisor (GCD) of two integers.
Greatest common divisor is also known as greatest common factor (gcf) and greatest common measure.
- Related task
- See also
-
- MathWorld entry: greatest common divisor.
- Wikipedia entry: greatest common divisor.
11l
F gcd(=u, =v)
L v != 0
(u, v) = (v, u % v)
R abs(u)
print(gcd(0, 0))
print(gcd(0, 10))
print(gcd(0, -10))
print(gcd(9, 6))
print(gcd(6, 9))
print(gcd(-6, 9))
print(gcd(8, 45))
print(gcd(40902, 24140))- Output:
0 10 10 3 3 3 1 34
360 Assembly
For maximum compatibility, this program uses only the basic instruction set (S/360) with 2 ASSIST macros (XDECO,XPRNT).
* Greatest common divisor 04/05/2016
GCD CSECT
USING GCD,R15 use calling register
L R6,A u=a
L R7,B v=b
LOOPW LTR R7,R7 while v<>0
BZ ELOOPW leave while
LR R8,R6 t=u
LR R6,R7 u=v
LR R4,R8 t
SRDA R4,32 shift to next reg
DR R4,R7 t/v
LR R7,R4 v=mod(t,v)
B LOOPW end while
ELOOPW LPR R9,R6 c=abs(u)
L R1,A a
XDECO R1,XDEC edit a
MVC PG+4(5),XDEC+7 move a to buffer
L R1,B b
XDECO R1,XDEC edit b
MVC PG+10(5),XDEC+7 move b to buffer
XDECO R9,XDEC edit c
MVC PG+17(5),XDEC+7 move c to buffer
XPRNT PG,80 print buffer
XR R15,R15 return code =0
BR R14 return to caller
A DC F'1071' a
B DC F'1029' b
PG DC CL80'gcd(00000,00000)=00000' buffer
XDEC DS CL12 temp for edit
YREGS
END GCD- Output:
gcd( 1071, 1029)= 21
PowerPC Assembly
Compile with:
gcc -mbig -mregnames -nostartfiles -nodefaultlibs -o gcd gcd.S
#include <syscall.h>
_gcd_string:
.ascii "gcd("
_gcd_string_len = . - _gcd_string
_gcd_close_string:
.ascii ") = "
_gcd_close_string_len = . - _gcd_close_string
.equiv STDIN, 0
.equiv STDOUT, 1
.align 4
.section ".text"
.global _start
.section ".opd","aw"
.align 3
_start:
.quad ._start,.TOC.@tocbase,0
.previous
.global ._start
._start:
li r30, 1071
li r31, 1029
# move the loaded values into the argument registers
mr r3, r30
mr r4, r31
bl gcd
# save the result for printing later
mr r29, r3
addis r4, r2, _gcd_string@toc@ha
addi r4, r4, _gcd_string@toc@l
li r5, _gcd_string_len
bl print_string
mr r3, r30
bl print_integer
li r3, ','
bl print_char
mr r3, r31
bl print_integer
addis r4, r2, _gcd_close_string@toc@ha
addi r4, r4, _gcd_close_string@toc@l
li r5, _gcd_close_string_len
bl print_string
mr r3, r29
bl print_integer
li r3, '\n'
bl print_char
li r0,SYS_exit # syscall number (sys_exit)
li r3,0 # first argument: exit code
sc # call kernel
gcd:
cmpd r3, r4
bge _gcd
mr r5, r3
mr r3, r4
mr r4, r5
_gcd:
cmpdi r4, 0
beqlr
mr r5, r3
mr r3, r4
modud r4, r5, r4
b _gcd
# r4 is the address of the string
# r5 is the length of the string
print_string:
li r0, SYS_write
li r3, STDOUT
sc
blr
print_char:
mr r4, r3
li r0, SYS_write
li r3, STDOUT
stb r4, -1(sp)
addi r4, sp, -1
li r5, 1
sc
blr
print_integer:
# r3 is the integer to print
# r4 is the working register
# r5 holds the current address to write to the string
# r6 is 10 for division operations
# r7 is working storage
mr r5, sp
li r6, 10
neg r4, r3
cmpdi r3, 0
bne 1f
li r7, '0'
stbu r7, -1(r5)
b 3f
1:
isellt r4, r4, r3 # r4 = abs(r3)
1:
modsd r7, r4, r6
divd r4, r4, r6
addi r7, r7, '0'
stbu r7, -1(r5)
cmpdi r4, 0
beq 1f
b 1b
1:
cmpdi r3, 0
blt 2f
3:
mr r4, r5
subf r5, r5, sp
mflr r14
bl print_string
mtlr r14
blr
2:
li r7, '-'
stbu r7, -1(r5)
b 3b
- Output:
gcd(1071,1029) = 21
8th
: gcd \ a b -- gcd
dup 0 n:= if drop ;; then
tuck \ b a b
n:mod \ b a-mod-b
recurse ;
: demo \ a b --
2dup "GCD of " . . " and " . . " = " . gcd . ;
100 5 demo cr
5 100 demo cr
7 23 demo cr
bye
- Output:
GCD of 5 and 100 = 5 GCD of 100 and 5 = 5 GCD of 23 and 7 = 1
AArch64 Assembly
/* ARM assembly AARCH64 Raspberry PI 3B */
/* program calPgcd64.s */
/*******************************************/
/* Constantes file */
/*******************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"
/*********************************/
/* Initialized data */
/*********************************/
.data
sMessResult: .asciz "Number 1 : @ number 2 : @ PGCD : @ \n"
szCarriageReturn: .asciz "\n"
szMessError: .asciz "Error PGCD !!\n"
/*********************************/
/* UnInitialized data */
/*********************************/
.bss
sZoneConv: .skip 24
/*********************************/
/* code section */
/*********************************/
.text
.global main
main: // entry of program
mov x20,36
mov x21,18
mov x0,x20
mov x1,x21
bl calPGCDmod
bcs 99f // error ?
mov x2,x0 // pgcd
mov x0,x20
mov x1,x21
bl displayResult
mov x20,37
mov x21,15
mov x0,x20
mov x1,x21
bl calPGCDmod
bcs 99f // error ?
mov x2,x0 // pgcd
mov x0,x20
mov x1,x21
bl displayResult
b 100f
99: // display error
ldr x0,qAdrszMessError
bl affichageMess
100: // standard end of the program
mov x0, #0 // return code
mov x8, #EXIT // request to exit program
svc #0 // perform the system call
qAdrszCarriageReturn: .quad szCarriageReturn
qAdrszMessError: .quad szMessError
/***************************************************/
/* Compute pgcd modulo use */
/***************************************************/
/* x0 contains first number */
/* x1 contains second number */
/* x0 return PGCD */
/* if error carry set to 1 */
calPGCDmod:
stp x1,lr,[sp,-16]! // save registres
stp x2,x3,[sp,-16]! // save registres
cbz x0,99f // if = 0 error
cbz x1,99f
cmp x0,0
bgt 1f
neg x0,x0 // if negative inversion number 1
1:
cmp x1,0
bgt 2f
neg x1,x1 // if negative inversion number 2
2:
cmp x0,x1 // compare two numbers
bgt 3f
mov x2,x0 // inversion
mov x0,x1
mov x1,x2
3:
udiv x2,x0,x1 // division
msub x0,x2,x1,x0 // compute remainder
cmp x0,0
bgt 2b // loop
mov x0,x1
cmn x0,0 // clear carry
b 100f
99: // error
mov x0,0
cmp x0,0 // set carry
100:
ldp x2,x3,[sp],16 // restaur des 2 registres
ldp x1,lr,[sp],16 // restaur des 2 registres
ret // retour adresse lr x30
/***************************************************/
/* display result */
/***************************************************/
/* x0 contains first number */
/* x1 contains second number */
/* x2 contains PGCD */
displayResult:
stp x1,lr,[sp,-16]! // save registres
mov x3,x1 // save x1
ldr x1,qAdrsZoneConv
bl conversion10 // décimal conversion
ldr x0,qAdrsMessResult
ldr x1,qAdrsZoneConv // insert conversion
bl strInsertAtCharInc
mov x4,x0 // save message address
mov x0,x3 // conversion second number
ldr x1,qAdrsZoneConv
bl conversion10 // décimal conversion
mov x0,x4 // move message address
ldr x1,qAdrsZoneConv // insert conversion
bl strInsertAtCharInc
mov x4,x0 // save message address
mov x0,x2 // conversion pgcd
ldr x1,qAdrsZoneConv
bl conversion10 // décimal conversion
mov x0,x4 // move message address
ldr x1,qAdrsZoneConv // insert conversion
bl strInsertAtCharInc
bl affichageMess // display message
ldp x1,lr,[sp],16 // restaur des 2 registres
ret // retour adresse lr x30
qAdrsMessResult: .quad sMessResult
qAdrsZoneConv: .quad sZoneConv
/********************************************************/
/* File Include fonctions */
/********************************************************/
/* for this file see task include a file in language AArch64 assembly */
.include "../includeARM64.inc"ACL2
(include-book "arithmetic-3/floor-mod/floor-mod" :dir :system)
(defun gcd$ (x y)
(declare (xargs :guard (and (natp x) (natp y))))
(cond ((or (not (natp x)) (< y 0))
nil)
((zp y) x)
(t (gcd$ y (mod x y)))))
Action!
CARD FUNC Gcd(CARD a,b)
CARD tmp
IF a<b THEN
tmp=a a=b b=tmp
FI
WHILE b#0
DO
tmp=a MOD b
a=b
b=tmp
OD
RETURN(a)
PROC Test(CARD a,b)
CARD res
res=Gcd(a,b)
PrintF("GCD of %I and %I is %I%E",a,b,res)
RETURN
PROC Main()
Test(48,18)
Test(9360,12240)
Test(17,19)
Test(123,1)
Test(0,0)
RETURN- Output:
Screenshot from Atari 8-bit computer
GCD of 48 and 18 is 6 GCD of 9360 and 12240 is 720 GCD of 17 and 19 is 1 GCD of 123 and 1 is 1 GCD of 0 and 0 is 0
ActionScript
//Euclidean algorithm
function gcd(a:int,b:int):int
{
var tmp:int;
//Swap the numbers so a >= b
if(a < b)
{
tmp = a;
a = b;
b = tmp;
}
//Find the gcd
while(b != 0)
{
tmp = a % b;
a = b;
b = tmp;
}
return a;
}
Ada
with Ada.Text_Io; use Ada.Text_Io;
procedure Gcd_Test is
function Gcd (A, B : Integer) return Integer is
M : Integer := A;
N : Integer := B;
T : Integer;
begin
while N /= 0 loop
T := M;
M := N;
N := T mod N;
end loop;
return M;
end Gcd;
begin
Put_Line("GCD of 100, 5 is" & Integer'Image(Gcd(100, 5)));
Put_Line("GCD of 5, 100 is" & Integer'Image(Gcd(5, 100)));
Put_Line("GCD of 7, 23 is" & Integer'Image(Gcd(7, 23)));
end Gcd_Test;
Output:
GCD of 100, 5 is 5 GCD of 5, 100 is 5 GCD of 7, 23 is 1
Aime
o_integer(gcd(33, 77));
o_byte('\n');
o_integer(gcd(49865, 69811));
o_byte('\n');ALGOL 60
begin
comment Greatest common divisor - algol 60;
integer procedure gcd(m,n);
value m,n;
integer m,n;
begin
integer a,b;
a:=abs(m);
b:=abs(n);
if a=0 then gcd:=b
else begin
integer c,i;
for i:=a while b notequal 0 do begin
c:=b;
b:=a-(a div b)*b;
a:=c
end;
gcd:=a
end
end gcd;
outinteger(1,gcd(21,35))
end- Output:
7
ALGOL 68
PROC gcd = (INT a, b) INT: (
IF a = 0 THEN
b
ELIF b = 0 THEN
a
ELIF a > b THEN
gcd(b, a MOD b)
ELSE
gcd(a, b MOD a)
FI
);
test:(
INT a = 33, b = 77;
printf(($x"The gcd of"g" and "g" is "gl$,a,b,gcd(a,b)));
INT c = 49865, d = 69811;
printf(($x"The gcd of"g" and "g" is "gl$,c,d,gcd(c,d)))
)Output:
The gcd of +33 and +77 is +11 The gcd of +49865 and +69811 is +9973
ALGOL-M
BEGIN
% RETURN P MOD Q %
INTEGER FUNCTION MOD (P, Q);
INTEGER P, Q;
BEGIN
MOD := P - Q * (P / Q);
END;
% RETURN GREATEST COMMON DIVISOR OF X AND Y %
INTEGER FUNCTION GCD (X, Y);
INTEGER X, Y;
BEGIN
INTEGER R;
IF X < Y THEN
BEGIN
INTEGER TEMP;
TEMP := X;
X := Y;
Y := TEMP;
END;
WHILE (R := MOD(X, Y)) <> 0 DO
BEGIN
X := Y;
Y := R;
END;
GCD := Y;
END;
COMMENT - EXERCISE THE FUNCTION;
WRITE("THE GCD OF 21 AND 35 IS", GCD(21,35));
WRITE("THE GCD OF 23 AND 35 IS", GCD(23,35));
WRITE("THE GCD OF 1071 AND 1029 IS", GCD(1071,1029));
WRITE("THE GCD OF 3528 AND 3780 IS", GCD(3528,252));
END- Output:
THE GCD OF 21 AND 35 IS 7 THE GCD OF 23 AND 35 IS 1 THE GCD OF 1071 AND 1029 IS 21 THE GCD OF 3528 AND 3780 IS 252
ALGOL W
begin
% iterative Greatest Common Divisor routine %
integer procedure gcd ( integer value m, n ) ;
begin
integer a, b, newA;
a := abs( m );
b := abs( n );
while b not = 0 do begin
newA := b;
b := a rem b;
a := newA;
end;
a
end gcd ;
write( gcd( -21, 35 ) );
end.Alore
def gcd(a as Int, b as Int) as Int
while b != 0
a,b = b, a mod b
end
return Abs(a)
endAntLang
AntLang has a built-in gcd function.
gcd[33; 77]It is not recommended, but possible to implement it on your own.
/Unoptimized version
gcd':{a:x;b:y;last[{(0 eq a mod x) min (0 eq b mod x)} hfilter {1 + x} map range[a max b]]}APL
33 49865 ∨ 77 69811 11 9973
If you're interested in how you'd write GCD in Dyalog, if Dyalog didn't have a primitive for it, (i.e. using other algorithms mentioned on this page: iterative, recursive, binary recursive), see different ways to write GCD in Dyalog.
⌈/(^/0=A∘.|X)/A←⍳⌊/X←49865 69811 9973
AppleScript
By recursion:
-- gcd :: Int -> Int -> Int
on gcd(a, b)
if b ≠ 0 then
gcd(b, a mod b)
else
if a < 0 then
-a
else
a
end if
end if
end gcd
And just for the sake of it, the same thing iteratively:
on hcf(a, b)
repeat until (b = 0)
set x to a
set a to b
set b to x mod b
end repeat
if (a < 0) then return -a
return a
end hcf
Arendelle
< a , b >
( r , @a )
[ @r != 0 ,
( r , @a % @b )
{ @r != 0 ,
( a , @b )
( b , @r )
}
]
( return , @b )
Arturo
print gcd [10 15]
- Output:
5
ATS
Stein’s algorithm, without proofs
Here is an implementation of Stein’s algorithm, without proofs of termination or correctness.
(********************************************************************)
(*
GCD of two integers, by Stein’s algorithm:
https://en.wikipedia.org/w/index.php?title=Binary_GCD_algorithm&oldid=1072393147
This is an implementation without proofs of anything.
The implementations shown here require the GCC builtin functions
for ‘count trailing zeros’. If your C compiler is GCC or another
that supports those functions, you are fine. Otherwise, one could
easily substitute other C code.
Compile with ‘patscc -o gcd gcd.dats’.
*)
#define ATS_EXTERN_PREFIX "rosettacode_gcd_"
#define ATS_DYNLOADFLAG 0 (* No initialization is needed. *)
#include "share/atspre_define.hats"
#include "share/atspre_staload.hats"
(********************************************************************)
(* *)
(* Declarations of the functions. *)
(* *)
(* g0uint_gcd_stein will be the generic template function for
unsigned integers. *)
extern fun {tk : tkind}
g0uint_gcd_stein :
(g0uint tk, g0uint tk) -<> g0uint tk
(* g0int_gcd_stein will be the generic template function for
signed integers, giving an unsigned result. *)
extern fun {tk_signed, tk_unsigned : tkind}
g0int_gcd_stein :
(g0int tk_signed, g0int tk_signed) -<> g0uint tk_unsigned
(* Let us call these functions ‘gcd_stein’ or just ‘gcd’. *)
overload gcd_stein with g0uint_gcd_stein
overload gcd_stein with g0int_gcd_stein
overload gcd with gcd_stein
(********************************************************************)
(* *)
(* The implementations. *)
(* *)
%{^
/*
We will need a ‘count trailing zeros of a positive number’ function,
but this is not provided in the ATS prelude. Here are
implementations using GCC builtin functions. For fast alternatives
in standard C, see
https://www.chessprogramming.org/index.php?title=BitScan&oldid=22495#Trailing_Zero_Count
*/
ATSinline() atstype_uint
rosettacode_gcd_g0uint_ctz_uint (atstype_uint x)
{
return __builtin_ctz (x);
}
ATSinline() atstype_ulint
rosettacode_gcd_g0uint_ctz_ulint (atstype_ulint x)
{
return __builtin_ctzl (x);
}
ATSinline() atstype_ullint
rosettacode_gcd_g0uint_ctz_ullint (atstype_ullint x)
{
return __builtin_ctzll (x);
}
%}
extern fun g0uint_ctz_uint : uint -<> int = "mac#%"
extern fun g0uint_ctz_ulint : ulint -<> int = "mac#%"
extern fun g0uint_ctz_ullint : ullint -<> int = "mac#%"
(* A generic template function for ‘count trailing zeros’ of
non-dependent unsigned integers. *)
extern fun {tk : tkind} g0uint_ctz : g0uint tk -<> int
(* Link the implementations to the template function. *)
implement g0uint_ctz<uint_kind> (x) = g0uint_ctz_uint x
implement g0uint_ctz<ulint_kind> (x) = g0uint_ctz_ulint x
implement g0uint_ctz<ullint_kind> (x) = g0uint_ctz_ullint x
(* Let one call the function simply ‘ctz’. *)
overload ctz with g0uint_ctz
(* Now the actual implementation of g0uint_gcd_stein, the template
function for the gcd of two unsigned integers. *)
implement {tk}
g0uint_gcd_stein (u, v) =
let
(* Make ‘t’ a shorthand for the unsigned integer type. *)
typedef t = g0uint tk
(* Use this macro to fake proof that an int is non-negative. *)
macdef nonneg (n) = $UNSAFE.cast{intGte 0} ,(n)
(* Looping is done by tail recursion. There is no proof
the function terminates; this fact is indicated by
‘<!ntm>’. *)
fun {tk : tkind}
main_loop (x_odd : t, y : t) :<!ntm> t =
let
(* Remove twos from y, giving an odd number.
Note gcd(x_odd,y_odd) = gcd(x_odd,y). *)
val y_odd = (y >> nonneg (ctz y))
in
if x_odd = y_odd then
x_odd
else
let
(* If y_odd < x_odd then swap x_odd and y_odd.
This operation does not affect the gcd. *)
val x_odd = min (x_odd, y_odd)
and y_odd = max (x_odd, y_odd)
in
main_loop (x_odd, y_odd - x_odd)
end
end
fn
u_and_v_both_positive (u : t, v : t) :<> t =
let
(* n = the number of common factors of two in u and v. *)
val n = ctz (u lor v)
(* Remove the common twos from u and v, giving x and y. *)
val x = (u >> nonneg n)
val y = (v >> nonneg n)
(* Remove twos from x, giving an odd number.
Note gcd(x_odd,y) = gcd(x,y). *)
val x_odd = (x >> nonneg (ctz x))
(* Run the main loop, but pretend it is proven to
terminate. Otherwise we could not write ‘<>’ above,
telling the ATS compiler that we trust the function
to terminate. *)
val z = $effmask_ntm (main_loop (x_odd, y))
in
(* Put the common factors of two back in. *)
(z << nonneg n)
end
(* If v < u then swap u and v. This operation does not
affect the gcd. *)
val u = min (u, v)
and v = max (u, v)
in
if iseqz u then
v
else
u_and_v_both_positive (u, v)
end
(* The implementation of g0int_gcd_stein, the template function for
the gcd of two signed integers, giving an unsigned result. *)
implement {signed_tk, unsigned_tk}
g0int_gcd_stein (u, v) =
let
val abs_u = $UNSAFE.cast{g0uint unsigned_tk} (abs u)
val abs_v = $UNSAFE.cast{g0uint unsigned_tk} (abs v)
in
g0uint_gcd_stein<unsigned_tk> (abs_u, abs_v)
end
(********************************************************************)
(* A demonstration program. *)
implement
main0 () =
begin
(* Unsigned integers. *)
assertloc (gcd (0U, 10U) = 10U);
assertloc (gcd (9UL, 6UL) = 3UL);
assertloc (gcd (40902ULL, 24140ULL) = 34ULL);
(* Signed integers. *)
assertloc (gcd (0, 10) = gcd (0U, 10U));
assertloc (gcd (~10, 0) = gcd (0U, 10U));
assertloc (gcd (~6L, ~9L) = 3UL);
assertloc (gcd (40902LL, 24140LL) = 34ULL);
assertloc (gcd (40902LL, ~24140LL) = 34ULL);
assertloc (gcd (~40902LL, 24140LL) = 34ULL);
assertloc (gcd (~40902LL, ~24140LL) = 34ULL);
assertloc (gcd (24140LL, 40902LL) = 34ULL);
assertloc (gcd (~24140LL, 40902LL) = 34ULL);
assertloc (gcd (24140LL, ~40902LL) = 34ULL);
assertloc (gcd (~24140LL, ~40902LL) = 34ULL)
end
(********************************************************************)
Stein’s algorithm, with proof of termination
Here is an implementation of Stein’s algorithm, this time with a proof of termination. Notice that the proof is rather ‘informal’; this is practical systems programming, not an exercise in mathematical logic.
(********************************************************************)
(*
GCD of two integers, by Stein’s algorithm:
https://en.wikipedia.org/w/index.php?title=Binary_GCD_algorithm&oldid=1072393147
This is an implementation with proof of termination.
The implementations shown here require the GCC builtin functions
for ‘count trailing zeros’. If your C compiler is GCC or another
that supports those functions, you are fine. Otherwise, one could
easily substitute other C code.
Compile with ‘patscc -o gcd gcd.dats’.
*)
#define ATS_EXTERN_PREFIX "rosettacode_gcd_"
#define ATS_DYNLOADFLAG 0 (* No initialization is needed. *)
#include "share/atspre_define.hats"
#include "share/atspre_staload.hats"
(********************************************************************)
(* *)
(* Declarations of the functions. *)
(* *)
(* g1uint_gcd_stein will be the generic template function for
unsigned integers. *)
extern fun {tk : tkind}
g0uint_gcd_stein :
(g0uint tk, g0uint tk) -<> g0uint tk
(* g0int_gcd_stein will be the generic template function for
signed integers, giving an unsigned result. *)
extern fun {tk_signed, tk_unsigned : tkind}
g0int_gcd_stein :
(g0int tk_signed, g0int tk_signed) -<> g0uint tk_unsigned
(* Let us call these functions ‘gcd_stein’ or just ‘gcd’. *)
overload gcd_stein with g0uint_gcd_stein
overload gcd_stein with g0int_gcd_stein
overload gcd with gcd_stein
(********************************************************************)
(* *)
(* The implementations. *)
(* *)
%{^
/*
We will need a ‘count trailing zeros of a positive number’ function,
but this is not provided in the ATS prelude. Here are
implementations using GCC builtin functions. For fast alternatives
in standard C, see
https://www.chessprogramming.org/index.php?title=BitScan&oldid=22495#Trailing_Zero_Count
*/
ATSinline() atstype_uint
rosettacode_gcd_g0uint_ctz_uint (atstype_uint x)
{
return __builtin_ctz (x);
}
ATSinline() atstype_ulint
rosettacode_gcd_g0uint_ctz_ulint (atstype_ulint x)
{
return __builtin_ctzl (x);
}
ATSinline() atstype_ullint
rosettacode_gcd_g0uint_ctz_ullint (atstype_ullint x)
{
return __builtin_ctzll (x);
}
%}
extern fun g0uint_ctz_uint : uint -<> int = "mac#%"
extern fun g0uint_ctz_ulint : ulint -<> int = "mac#%"
extern fun g0uint_ctz_ullint : ullint -<> int = "mac#%"
(* A generic template function for ‘count trailing zeros’ of
non-dependent unsigned integers. *)
extern fun {tk : tkind} g0uint_ctz : g0uint tk -<> int
(* Link the implementations to the template function. *)
implement g0uint_ctz<uint_kind> (x) = g0uint_ctz_uint x
implement g0uint_ctz<ulint_kind> (x) = g0uint_ctz_ulint x
implement g0uint_ctz<ullint_kind> (x) = g0uint_ctz_ullint x
(* Let one call the function simply ‘ctz’. *)
overload ctz with g0uint_ctz
(* Now the actual implementation of g0uint_gcd_stein, the template
function for the gcd of two unsigned integers. *)
implement {tk}
g0uint_gcd_stein (u, v) =
let
(* Make ‘t’ a shorthand for the unsigned integer types. *)
typedef t = g0uint tk
typedef t (i : int) = g1uint (tk, i)
(* Use this macro to fake proof that an int is non-negative. *)
macdef nonneg (n) = $UNSAFE.cast{intGte 0} ,(n)
(* Looping is done by tail recursion. The value of (x_odd + y)
must decrease towards zero, to prove termination. *)
fun {tk : tkind}
main_loop {x_odd, y : pos} .<x_odd + y>.
(x_odd : t (x_odd), y : t (y)) :<>
[d : pos] t (d) =
let
(* Remove twos from y, giving an odd number. Note
gcd(x_odd,y_odd) = gcd(x_odd,y). We do not have a
dependent-type version of the following operation, so let
us do it with non-dependent types. *)
val [y_odd : int] y_odd =
g1ofg0 ((g0ofg1 y) >> nonneg (ctz (g0ofg1 y)))
(* Assert some things we know without proof. (You could also
use assertloc, which inserts a runtime check.) *)
prval _ = $UNSAFE.prop_assert {0 < y_odd} ()
prval _ = $UNSAFE.prop_assert {y_odd <= y} ()
in
if x_odd = y_odd then
x_odd
else if y_odd < x_odd then
main_loop (y_odd, x_odd - y_odd)
else
main_loop (x_odd, y_odd - x_odd)
end
fn
u_and_v_both_positive (u : t, v : t) :<> t =
let
(* n = the number of common factors of two in u and v. *)
val n = ctz (u lor v)
(* Remove the common twos from u and v, giving x and y. *)
val x = (u >> nonneg n)
val y = (v >> nonneg n)
(* Remove twos from x, giving an odd number.
Note gcd(x_odd,y) = gcd(x,y). *)
val x_odd = (x >> nonneg (ctz x))
(* To prove termination of main_loop, we have to
convert x_odd and y to a dependent type. *)
val [x_odd : int] x_odd = g1ofg0 x_odd
val [y : int] y = g1ofg0 y
(* Assert that they are positive. (One could also use
assertloc, which inserts a runtime check.) *)
prval _ = $UNSAFE.prop_assert {0 < x_odd} ()
prval _ = $UNSAFE.prop_assert {0 < y} ()
val z = main_loop (x_odd, y)
(* Convert back to the non-dependent type. *)
val z = g0ofg1 z
in
(* Put the common factors of two back in. *)
(z << nonneg n)
end
(* If v < u then swap u and v. This operation does not
affect the gcd. *)
val u = min (u, v)
and v = max (u, v)
in
if iseqz u then
v
else
u_and_v_both_positive (u, v)
end
(* The implementation of g0int_gcd_stein, the template function for
the gcd of two signed integers, giving an unsigned result. *)
implement {signed_tk, unsigned_tk}
g0int_gcd_stein (u, v) =
let
val abs_u = $UNSAFE.cast{g0uint unsigned_tk} (abs u)
val abs_v = $UNSAFE.cast{g0uint unsigned_tk} (abs v)
in
g0uint_gcd_stein<unsigned_tk> (abs_u, abs_v)
end
(********************************************************************)
(* A demonstration program. *)
implement
main0 () =
begin
(* Unsigned integers. *)
assertloc (gcd (0U, 10U) = 10U);
assertloc (gcd (9UL, 6UL) = 3UL);
assertloc (gcd (40902ULL, 24140ULL) = 34ULL);
(* Signed integers. *)
assertloc (gcd (0, 10) = gcd (0U, 10U));
assertloc (gcd (~10, 0) = gcd (0U, 10U));
assertloc (gcd (~6L, ~9L) = 3UL);
assertloc (gcd (40902LL, 24140LL) = 34ULL);
assertloc (gcd (40902LL, ~24140LL) = 34ULL);
assertloc (gcd (~40902LL, 24140LL) = 34ULL);
assertloc (gcd (~40902LL, ~24140LL) = 34ULL);
assertloc (gcd (24140LL, 40902LL) = 34ULL);
assertloc (gcd (~24140LL, 40902LL) = 34ULL);
assertloc (gcd (24140LL, ~40902LL) = 34ULL);
assertloc (gcd (~24140LL, ~40902LL) = 34ULL)
end
(********************************************************************)Euclid’s algorithm, with proof of termination and correctness
Here is an implementation of Euclid’s algorithm, with a proof of correctness.
(********************************************************************)
(*
GCD of two integers, by Euclid’s algorithm; verified with proofs.
Compile with ‘patscc -o gcd gcd.dats’.
*)
#define ATS_DYNLOADFLAG 0 (* No initialization is needed. *)
#include "share/atspre_define.hats"
#include "share/atspre_staload.hats"
(********************************************************************)
(* *)
(* Definition of the gcd by axioms in the static language. *)
(* *)
(* (‘Props’ are better supported in ATS, but I enjoy using the *)
(* the static language in proofs.) *)
(* *)
(* Write the gcd as an undefined static function. It will be defined
implicitly by axioms. (Such a function also can be used with an
external SMT solver such as CVC4, but using an external solver is
not the topic of this program.) *)
stacst gcd (u : int, v : int) : int
(*
I think the reader will accept the following axioms as valid,
if gcd(0, 0) is to be defined as equal to zero.
(An exercise for the reader is to prove ‘gcd_of_remainder’
from gcd (u, v) == gcd (u, v - u). This requires definitions
of multiplication and Euclidean division, which are encoded
in terms of props in ‘prelude/SATS/arith_prf.sats’.)
*)
extern praxi
gcd_of_zero :
{u, v : int | u == 0; 0 <= v} (* For all integers u = 0,
v non-negative. *)
() -<prf> [gcd (u, v) == v] void
extern praxi
gcd_of_remainder :
{u, v : int | 0 < u; 0 <= v} (* For all integers u positive,
v non-negative. *)
() -<prf> [gcd (u, v) == gcd (u, v mod u)] void
extern praxi
gcd_is_commutative :
{u, v : int} (* For all integers u, v. *)
() -<prf> [gcd (u, v) == gcd (v, u)] void
extern praxi
gcd_of_the_absolute_values :
{u, v : int} (* For all integers u, v. *)
() -<prf> [gcd (u, v) == gcd (abs u, abs v)] void
extern praxi
gcd_is_a_function :
{u1, v1 : int}
{u2, v2 : int | u1 == u2; v1 == v2}
() -<prf> [gcd (u1, v1) == gcd (u2, v2)] void
(********************************************************************)
(* *)
(* Function declarations. *)
(* *)
(* g1uint_gcd_euclid will be the generic template function for
unsigned integers. *)
extern fun {tk : tkind}
g1uint_gcd_euclid :
{u, v : int}
(g1uint (tk, u),
g1uint (tk, v)) -<>
g1uint (tk, gcd (u, v))
(* g1int_gcd_euclid will be the generic template function for
signed integers, giving an unsigned result. *)
extern fun {tk_signed, tk_unsigned : tkind}
g1int_gcd_euclid :
{u, v : int}
(g1int (tk_signed, u),
g1int (tk_signed, v)) -<>
g1uint (tk_unsigned, gcd (u, v))
(* Let us call these functions ‘gcd_euclid’ or just ‘gcd’. *)
overload gcd_euclid with g1uint_gcd_euclid
overload gcd_euclid with g1int_gcd_euclid
overload gcd with gcd_euclid
(********************************************************************)
(* *)
(* Function implementations. *)
(* *)
(* The implementation of the remainder function in the ATS2 prelude
is inconvenient for us; it does not say that the result equals
‘u mod v’. Let us reimplement it more to our liking. *)
fn {tk : tkind}
g1uint_rem {u, v : int | v != 0}
(u : g1uint (tk, u),
v : g1uint (tk, v)) :<>
[w : int | 0 <= w; w < v; w == u mod v]
g1uint (tk, w) =
let
prval _ = lemma_g1uint_param u
prval _ = lemma_g1uint_param v
in
$UNSAFE.cast (g1uint_mod (u, v))
end
implement {tk}
g1uint_gcd_euclid {u, v} (u, v) =
let
(* The static variable v, which is defined within the curly
braces, must, with each iteration, approach zero without
passing it. Otherwise the loop is not proven to terminate,
and the typechecker will reject it. *)
fun
loop {u, v : int | 0 <= u; 0 <= v} .<v>.
(u : g1uint (tk, u),
v : g1uint (tk, v)) :<>
g1uint (tk, gcd (u, v)) =
if v = g1i2u 0 then
let
(* prop_verify tests whether what we believe we have
proven has actually been proven. Using it a lot lengthens
the code but is excellent documentation. *)
prval _ = prop_verify {0 <= u} ()
prval _ = prop_verify {v == 0} ()
prval _ = gcd_of_zero {v, u} ()
prval _ = prop_verify {gcd (v, u) == u} ()
prval _ = gcd_is_commutative {u, v} ()
prval _ = prop_verify {gcd (u, v) == gcd (v, u)} ()
(* Therefore, by transitivity of equality: *)
prval _ = prop_verify {gcd (u, v) == u} ()
in
u
end
else
let
prval _ = prop_verify {0 <= u} ()
prval _ = prop_verify {0 < v} ()
prval _ = gcd_of_remainder {v, u} ()
prval _ = prop_verify {gcd (v, u) == gcd (v, u mod v)} ()
prval _ = gcd_is_commutative {u, v} ()
prval _ = prop_verify {gcd (u, v) == gcd (v, u)} ()
(* Therefore, by transitivity of equality: *)
prval _ = prop_verify {gcd (u, v) == gcd (v, u mod v)} ()
val [w : int] w = g1uint_rem (u, v)
prval _ = prop_verify {0 <= w} ()
prval _ = prop_verify {w < v} ()
prval _ = prop_verify {w == u mod v} ()
(* It has been proven that the function will terminate: *)
prval _ = prop_verify {0 <= w && w < v} ()
prval _ = gcd_is_a_function {v, u mod v} {v, w} ()
prval _ = prop_verify {gcd (v, u mod v) == gcd (v, w)} ()
(* Therefore, by transitivity of equality: *)
prval _ = prop_verify {gcd (u, v) == gcd (v, w)} ()
in
loop (v, w)
end
(* u is unsigned, thus proving 0 <= u. *)
prval _ = lemma_g1uint_param (u)
(* v is unsigned, thus proving 0 <= v. *)
prval _ = lemma_g1uint_param (v)
in
loop (u, v)
end
implement {tk_signed, tk_unsigned}
g1int_gcd_euclid {u, v} (u, v) =
let
(* Prove that gcd(abs u, abs v) equals gcd(u, v). *)
prval _ = gcd_of_the_absolute_values {u, v} ()
in
(* Compute gcd(abs u, abs v). The ‘g1i2u’ notations cast the
values from signed integers to unsigned integers. *)
g1uint_gcd_euclid (g1i2u (abs u), g1i2u (abs v))
end
(********************************************************************)
(* *)
(* A demonstration program. *)
(* *)
(* Unfortunately, the ATS prelude may not include implementations *)
(* of all the functions we need for long and long long integers. *)
(* Thus the demonstration will be entirely on regular int and uint. *)
(* *)
(* (Including implementations here would distract from the purpose. *)
(* *)
implement
main0 () =
begin
(* Unsigned integers. *)
assertloc (gcd (0U, 10U) = 10U);
assertloc (gcd (9U, 6U) = 3U);
assertloc (gcd (40902U, 24140U) = 34U);
(* Signed integers. *)
assertloc (gcd (0, 10) = gcd (0U, 10U));
assertloc (gcd (~10, 0) = gcd (0U, 10U));
assertloc (gcd (~6, ~9) = 3U);
assertloc (gcd (40902, 24140) = 34U);
assertloc (gcd (40902, ~24140) = 34U);
assertloc (gcd (~40902, 24140) = 34U);
assertloc (gcd (~40902, ~24140) = 34U);
assertloc (gcd (24140, 40902) = 34U);
assertloc (gcd (~24140, 40902) = 34U);
assertloc (gcd (24140, ~40902) = 34U);
assertloc (gcd (~24140, ~40902) = 34U)
end
(********************************************************************)Some proofs about the gcd
For the sake of interest, here is some use of ATS's "props"-based proof system. There is no executable code in the following.
(* Typecheck this file with ‘patscc -tcats gcd-proofs.dats’. *)
(* Definition of the gcd by Euclid’s algorithm, using subtractions;
gcd(0,0) is defined to equal zero. (I do not prove that this
definition is equivalent to the common meaning of ‘greatest common
divisor’; that’s not a sort of thing ATS is good at.) *)
dataprop GCD (int, int, int) =
| GCD_0_0 (0, 0, 0)
| {u : pos}
GCD_u_0 (u, 0, u)
| {v : pos}
GCD_0_v (0, v, v)
| {u, v : pos | u <= v}
{d : pos}
GCD_u_le_v (u, v, d) of
GCD (u, v - u, d)
| {u, v : pos | u > v}
{d : pos}
GCD_u_gt_v (u, v, d) of
GCD (u - v, v, d)
| {u, v : int | u < 0 || v < 0}
{d : pos}
GCD_u_or_v_neg (u, v, d) of
GCD (abs u, abs v, d)
(* Here is a proof, by construction, of the proposition
‘The gcd of 12 and 8 is 4’. *)
prfn
gcd_12_8 () :<prf>
GCD (12, 8, 4) =
let
prval pf = GCD_u_0 {4} ()
prval pf = GCD_u_le_v {4, 4} {4} (pf)
prval pf = GCD_u_le_v {4, 8} {4} (pf)
prval pf = GCD_u_gt_v {12, 8} {4} (pf)
in
pf
end
(* A lemma: the gcd is total. That is, it is defined for all
integers. *)
extern prfun
gcd_istot :
{u, v : int}
() -<prf>
[d : int]
GCD (u, v, d)
(* Another lemma: the gcd is a function: it has a unique value for
any given pair of arguments. *)
extern prfun
gcd_isfun :
{u, v : int}
{d, e : int}
(GCD (u, v, d),
GCD (u, v, e)) -<prf>
[d == e] void
(* Proof of gcd_istot. This source file will not pass typechecking
unless the proof is valid. *)
primplement
gcd_istot {u, v} () =
let
prfun
gcd_istot__nat_nat__ {u, v : nat | u != 0 || v != 0} .<u + v>.
() :<prf> [d : pos] GCD (u, v, d) =
sif v == 0 then
GCD_u_0 ()
else sif u == 0 then
GCD_0_v ()
else sif u <= v then
GCD_u_le_v (gcd_istot__nat_nat__ {u, v - u} ())
else
GCD_u_gt_v (gcd_istot__nat_nat__ {u - v, v} ())
prfun
gcd_istot__int_int__ {u, v : int | u != 0 || v != 0} .<>.
() :<prf> [d : pos] GCD (u, v, d) =
sif u < 0 || v < 0 then
GCD_u_or_v_neg (gcd_istot__nat_nat__ {abs u, abs v} ())
else
gcd_istot__nat_nat__ {u, v} ()
in
sif u == 0 && v == 0 then
GCD_0_0 ()
else
gcd_istot__int_int__ {u, v} ()
end
(* Proof of gcd_isfun. This source file will not pass typechecking
unless the proof is valid. *)
primplement
gcd_isfun {u, v} {d, e} (pfd, pfe) =
let
prfun
gcd_isfun__nat_nat__ {u, v : nat}
{d, e : int}
.<u + v>.
(pfd : GCD (u, v, d),
pfe : GCD (u, v, e)) :<prf> [d == e] void =
case+ pfd of
| GCD_0_0 () =>
{
prval GCD_0_0 () = pfe
}
| GCD_u_0 () =>
{
prval GCD_u_0 () = pfe
}
| GCD_0_v () =>
{
prval GCD_0_v () = pfe
}
| GCD_u_le_v pfd1 =>
{
prval GCD_u_le_v pfe1 = pfe
prval _ = gcd_isfun__nat_nat__ (pfd1, pfe1)
}
| GCD_u_gt_v pfd1 =>
{
prval GCD_u_gt_v pfe1 = pfe
prval _ = gcd_isfun__nat_nat__ (pfd1, pfe1)
}
in
sif u < 0 || v < 0 then
{
prval GCD_u_or_v_neg pfd1 = pfd
prval GCD_u_or_v_neg pfe1 = pfe
prval _ = gcd_isfun__nat_nat__ (pfd1, pfe1)
}
else
gcd_isfun__nat_nat__ (pfd, pfe)
endAutoHotkey
Contributed by Laszlo on the ahk forum
GCD(a,b) {
Return b=0 ? Abs(a) : Gcd(b,mod(a,b))
}
Significantly faster than recursion:
GCD(a, b) {
while b
b := Mod(a | 0x0, a := b)
return a
}
AutoIt
_GCD(18, 12)
_GCD(1071, 1029)
_GCD(3528, 3780)
Func _GCD($ia, $ib)
Local $ret = "GCD of " & $ia & " : " & $ib & " = "
Local $imod
While True
$imod = Mod($ia, $ib)
If $imod = 0 Then Return ConsoleWrite($ret & $ib & @CRLF)
$ia = $ib
$ib = $imod
WEnd
EndFunc ;==>_GCD
Output:
GCD of 18 : 12 = 6 GCD of 1071 : 1029 = 21 GCD of 3528 : 3780 = 252
AWK
The following scriptlet defines the gcd() function, then reads pairs of numbers from stdin, and reports their gcd on stdout.
$ awk 'function gcd(p,q){return(q?gcd(q,(p%q)):p)}{print gcd($1,$2)}'
12 16
4
22 33
11
45 67
1
Axe
Lbl GCD
r₁→A
r₂→B
!If B
A
Return
End
GCD(B,A^B)BASIC
Applesoft BASIC
0 A = ABS(INT(A))
1 B = ABS(INT(B))
2 GCD = A * NOT NOT B
3 FOR B = B + A * NOT B TO 0 STEP 0
4 A = GCD
5 GCD = B
6 B = A - INT (A / GCD) * GCD
7 NEXT BBASIC256
Iterative
function gcdI(x, y)
while y
t = y
y = x mod y
x = t
end while
return x
end function
# ------ test ------
a = 111111111111111
b = 11111
print : print "GCD(";a;", ";b;") = "; gcdI(a, b)
print : print "GCD(";a;", 111) = "; gcdI(a, 111)
end
- Output:
Igual que la entrada de FreeBASIC.
Recursive
function gcdp(a, b)
if b = 0 then return a
return gcdp(b, a mod b)
end function
function gcdR(a, b)
return gcdp(abs(a), abs(b))
end function
BBC BASIC
DEF FN_GCD_Iterative_Euclid(A%, B%)
LOCAL C%
WHILE B%
C% = A%
A% = B%
B% = C% MOD B%
ENDWHILE
= ABS(A%)
Chipmunk Basic
Iterative
100 cls
110 a = 40902
120 b = 24140
130 print : print "GCD(";a;", ";b;") = ";gcdi(a,b)
140 print : print "GCD(";a;", 111) = ";gcdi(a,111)
170 end
190 sub gcdi(x,y)
200 while y
210 t = y
220 y = x mod y
230 x = t
240 wend
250 gcdi = x
260 end sub
FreeBASIC
Iterative solution
' version 17-06-2015
' compile with: fbc -s console
Function gcd(x As ULongInt, y As ULongInt) As ULongInt
Dim As ULongInt t
While y
t = y
y = x Mod y
x = t
Wend
Return x
End Function
' ------=< MAIN >=------
Dim As ULongInt a = 111111111111111
Dim As ULongInt b = 11111
Print : Print "GCD(";a;", ";b;") = "; gcd(a, b)
Print : Print "GCD(";a;", 111) = "; gcd(a, 111)
' empty keyboard buffer
While InKey <> "" : Wend
Print : Print : Print "hit any key to end program"
Sleep
End
- Output:
GCD(111111111111111, 11111) = 11111 GCD(111111111111111, 111) = 111
Recursive solution
function gcdp( a as uinteger, b as uinteger ) as uinteger
if b = 0 then return a
return gcdp( b, a mod b )
end function
function gcd(a as integer, b as integer) as uinteger
return gcdp( abs(a), abs(b) )
end function
GFA Basic
'
' Greatest Common Divisor
'
a%=24
b%=112
PRINT "GCD of ";a%;" and ";b%;" is ";@gcd(a%,b%)
'
' Function computes gcd
'
FUNCTION gcd(a%,b%)
LOCAL t%
'
WHILE b%<>0
t%=a%
a%=b%
b%=t% MOD b%
WEND
'
RETURN ABS(a%)
ENDFUNC
GW-BASIC
10 INPUT A, B
20 IF A < 0 THEN A = -A
30 IF B < 0 THEN B = -B
40 GOTO 70
50 PRINT A
60 END
70 IF B = 0 THEN GOTO 50
80 TEMP = B
90 B = A MOD TEMP
100 A = TEMP
110 GOTO 70
IS-BASIC
100 DEF GCD(A,B)
110 DO WHILE B>0
120 LET T=B
130 LET B=MOD(A,B)
140 LET A=T
150 LOOP
160 LET GCD=A
170 END DEF
180 PRINT GCD(12,16)Liberty BASIC
'iterative Euclid algorithm
print GCD(-2,16)
end
function GCD(a,b)
while b
c = a
a = b
b = c mod b
wend
GCD = abs(a)
end functionPureBasic
Iterative
Import "" ;msvcrt.lib
AbsI(Quad.q) As "_abs64"
AbsL(Long.l) As "labs"
EndImport
Procedure.i GCD(u.i, v.i)
Protected.i t
While v <> 0
t = v
v = u % v
u = t
Wend
ProcedureReturn AbsI(u) ; Avoid float conversion with Abs(u).
EndProcedure
Debug GCD(18, 12) ; 6
Debug GCD(1071, 1029) ; 21
Debug GCD(3528, -3780) ; 252
QuickBASIC
Iterative
DECLARE FUNCTION gcd (a%, b%)
PRINT gcd(18, 30)
END
FUNCTION gcd (a%, b%)
WHILE b% <> 0
t% = b%
b% = a% MOD b%
a% = t%
WEND
gcd = ABS(a%)
END FUNCTION
- Output:
6
Recursive
DECLARE FUNCTION gcd (a%, b%)
PRINT gcd(30, 18)
END
FUNCTION gcd (a%, b%)
IF b% = 0 THEN
gcd = ABS(a%)
ELSE
gcd = gcd(b%, a% MOD b%)
END IF
END FUNCTION
- Output:
6
Run BASIC
print abs(gcd(-220,160))
function gcd(gcd,b)
while b
c = gcd
gcd = b
b = c mod b
wend
end functionS-BASIC
rem - return p mod q
function mod(p, q = integer) = integer
end = p - q * (p / q)
rem - return greatest common divisor of x and y
function gcd(x, y = integer) = integer
var r, temp = integer
if x < y then
begin
temp = x
x = y
y = temp
end
r = mod(x, y)
while r <> 0 do
begin
x = y
y = r
r = mod(x, y)
end
end = y
rem - exercise the function
print "The GCD of 21 and 35 is"; gcd(21,35)
print "The GCD of 23 and 35 is"; gcd(23,35)
print "The GCD of 1071 and 1029 is"; gcd(1071, 1029)
print "The GCD of 3528 and 3780 is"; gcd(3528,3780)
end
- Output:
The GCD of 21 and 35 is 7 The GCD of 23 and 35 is 1 The GCD of 1071 and 1029 is 21 The GCD of 3528 and 3780 is 252
Sinclair ZX81 BASIC
10 LET M=119
20 LET N=544
30 LET R=M-N*INT (M/N)
40 IF R=0 THEN GOTO 80
50 LET M=N
60 LET N=R
70 GOTO 30
80 PRINT N
- Output:
17
TI-83 BASIC , TI-89 BASIC
gcd(A,B)
The ) can be omitted in TI-83 basic
Tiny BASIC
10 PRINT "First number"
20 INPUT A
30 PRINT "Second number"
40 INPUT B
50 IF A<0 THEN LET A=-A
60 IF B<0 THEN LET B=-B
70 IF A>B THEN GOTO 130
80 LET B = B - A
90 IF A=0 THEN GOTO 110
100 GOTO 50
110 PRINT B
120 END
130 LET C=A
140 LET A=B
150 LET B=C
160 GOTO 70
True BASIC
REM Iterative solution
FUNCTION gcdI(x, y)
DO WHILE y > 0
LET t = y
LET y = remainder(x, y)
LET x = t
LOOP
LET gcdI = x
END FUNCTION
LET a = 111111111111111
LET b = 11111
PRINT
PRINT "GCD(";a;", ";b;") = "; gcdI(a, b)
PRINT
PRINT "GCD(";a;", 111) = "; gcdI(a, 111)
END
uBasic/4tH
Print "GCD of 18 : 12 = "; FUNC(_GCD_Iterative_Euclid(18,12))
Print "GCD of 1071 : 1029 = "; FUNC(_GCD_Iterative_Euclid(1071,1029))
Print "GCD of 3528 : 3780 = "; FUNC(_GCD_Iterative_Euclid(3528,3780))
End
_GCD_Iterative_Euclid Param(2)
Local (1)
Do While b@
c@ = a@
a@ = b@
b@ = c@ % b@
Loop
Return (Abs(a@))
- Output:
GCD of 18 : 12 = 6 GCD of 1071 : 1029 = 21 GCD of 3528 : 3780 = 252 0 OK, 0:205
VBA
Function gcd(u As Long, v As Long) As Long
Dim t As Long
Do While v
t = u
u = v
v = t Mod v
Loop
gcd = u
End Function
This function uses repeated subtractions. Simple but not very efficient.
Public Function GCD(a As Long, b As Long) As Long
While a <> b
If a > b Then a = a - b Else b = b - a
Wend
GCD = a
End Function- Output:
Example:
print GCD(1280, 240) 80 print GCD(3475689, 23566319) 7 a=123456789 b=234736437 print GCD((a),(b)) 3
A note on the last example: using brackets forces a and b to be evaluated before GCD is called. Not doing this will cause a compile error because a and b are not the same type as in the function declaration (they are Variant, not Long). Alternatively you can use the conversion function CLng as in print GCD(CLng(a),CLng(b))
VBScript
Function GCD(a,b)
Do
If a Mod b > 0 Then
c = a Mod b
a = b
b = c
Else
GCD = b
Exit Do
End If
Loop
End Function
WScript.Echo "The GCD of 48 and 18 is " & GCD(48,18) & "."
WScript.Echo "The GCD of 1280 and 240 is " & GCD(1280,240) & "."
WScript.Echo "The GCD of 1280 and 240 is " & GCD(3475689,23566319) & "."
WScript.Echo "The GCD of 1280 and 240 is " & GCD(123456789,234736437) & "."
- Output:
The GCD of 48 and 18 is 6. The GCD of 1280 and 240 is 80. The GCD of 1280 and 240 is 7. The GCD of 1280 and 240 is 3.
Visual Basic
Function GCD(ByVal a As Long, ByVal b As Long) As Long
Dim h As Long
If a Then
If b Then
Do
h = a Mod b
a = b
b = h
Loop While b
End If
GCD = Abs(a)
Else
GCD = Abs(b)
End If
End Function
Sub Main()
' testing the above function
Debug.Assert GCD(12, 18) = 6
Debug.Assert GCD(1280, 240) = 80
Debug.Assert GCD(240, 1280) = 80
Debug.Assert GCD(-240, 1280) = 80
Debug.Assert GCD(240, -1280) = 80
Debug.Assert GCD(0, 0) = 0
Debug.Assert GCD(0, 1) = 1
Debug.Assert GCD(1, 0) = 1
Debug.Assert GCD(3475689, 23566319) = 7
Debug.Assert GCD(123456789, 234736437) = 3
Debug.Assert GCD(3780, 3528) = 252
End Sub
XBasic
' Greatest common divisor
PROGRAM "gcddemo"
VERSION "0.001"
DECLARE FUNCTION Entry()
DECLARE FUNCTION GcdRecursive(u&, v&)
DECLARE FUNCTION GcdIterative(u&, v&)
DECLARE FUNCTION GcdBinary(u&, v&)
FUNCTION Entry()
m& = 49865
n& = 69811
PRINT "GCD("; LTRIM$(STR$(m&)); ","; n&; "):"; GcdIterative(m&, n&); " (iterative)"
PRINT "GCD("; LTRIM$(STR$(m&)); ","; n&; "):"; GcdRecursive(m&, n&); " (recursive)"
PRINT "GCD("; LTRIM$(STR$(m&)); ","; n&; "):"; GcdBinary (m&, n&); " (binary)"
END FUNCTION
FUNCTION GcdRecursive(u&, v&)
IF u& MOD v& <> 0 THEN
RETURN GcdRecursive(v&, u& MOD v&)
ELSE
RETURN v&
END IF
END FUNCTION
FUNCTION GcdIterative(u&, v&)
DO WHILE v& <> 0
t& = u&
u& = v&
v& = t& MOD v&
LOOP
RETURN ABS(u&)
END FUNCTION
FUNCTION GcdBinary(u&, v&)
u& = ABS(u&)
v& = ABS(v&)
IF u& < v& THEN
t& = u&
u& = v&
v& = t&
END IF
IF v& = 0 THEN
RETURN u&
ELSE
k& = 1
DO WHILE (u& MOD 2 = 0) && (v& MOD 2 = 0)
u& = u& >> 1
v& = v& >> 1
k& = k& << 1
LOOP
IF u& MOD 2 = 0 THEN
t& = u&
ELSE
t& = -v&
END IF
DO WHILE t& <> 0
DO WHILE t& MOD 2 = 0
t& = t& \ 2
LOOP
IF t& > 0 THEN
u& = t&
ELSE
v& = -t&
END IF
t& = u& - v&
LOOP
RETURN u& * k&
END IF
END FUNCTION
END PROGRAM
- Output:
GCD(49865, 69811): 9973 (iterative) GCD(49865, 69811): 9973 (recursive) GCD(49865, 69811): 9973 (binary)
Yabasic
sub gcd(u, v)
local t
u = int(abs(u))
v = int(abs(v))
while(v)
t = u
u = v
v = mod(t, v)
wend
return u
end sub
print "Greatest common divisor: ", gcd(12345, 9876)ZX Spectrum Basic
10 FOR n=1 TO 3
20 READ a,b
30 PRINT "GCD of ";a;" and ";b;" = ";
40 GO SUB 70
50 NEXT n
60 STOP
70 IF b=0 THEN PRINT ABS (a): RETURN
80 LET c=a: LET a=b: LET b=FN m(c,b): GO TO 70
90 DEF FN m(a,b)=a-INT (a/b)*b
100 DATA 12,16,22,33,45,67
Batch File
Recursive method
:: gcd.cmd
@echo off
:gcd
if "%2" equ "" goto :instructions
if "%1" equ "" goto :instructions
if %2 equ 0 (
set final=%1
goto :done
)
set /a res = %1 %% %2
call :gcd %2 %res%
goto :eof
:done
echo gcd=%final%
goto :eof
:instructions
echo Syntax:
echo GCD {a} {b}
echo.
Bc
Utility functions:
define even(a)
{
if ( a % 2 == 0 ) {
return(1);
} else {
return(0);
}
}
define abs(a)
{
if (a<0) {
return(-a);
} else {
return(a);
}
}
Iterative (Euclid)
define gcd_iter(u, v)
{
while(v) {
t = u;
u = v;
v = t % v;
}
return(abs(u));
}
Recursive
define gcd(u, v)
{
if (v) {
return ( gcd(v, u%v) );
} else {
return (abs(u));
}
}
Iterative (Binary)
define gcd_bin(u, v)
{
u = abs(u);
v = abs(v);
if ( u < v ) {
t = u; u = v; v = t;
}
if ( v == 0 ) { return(u); }
k = 1;
while (even(u) && even(v)) {
u = u / 2; v = v / 2;
k = k * 2;
}
if ( even(u) ) {
t = -v;
} else {
t = u;
}
while (t) {
while (even(t)) {
t = t / 2;
}
if (t > 0) {
u = t;
} else {
v = -t;
}
t = u - v;
}
return (u * k);
}
BCPL
get "libhdr"
let gcd(m,n) = n=0 -> m, gcd(n, m rem n)
let show(m,n) be
writef("gcd(%N, %N) = %N*N", m, n, gcd(m, n))
let start() be
$( show(18,12)
show(1071,1029)
show(3528,3780)
$)- Output:
gcd(18, 12) = 6 gcd(1071, 1029) = 21 gcd(3528, 3780) = 252
Befunge
#v&< @.$<
:<\g05%p05:_^#
BQN
Gcd ← {𝕨(|𝕊⍟(>⟜0)⊣)𝕩}
Example:
21 Gcd 35
7
Bracmat
Bracmat uses the Euclidean algorithm to simplify fractions. The den function extracts the denominator from a fraction.
(gcd=a b.!arg:(?a.?b)&!b*den$(!a*!b^-1)^-1);Example:
{?} gcd$(49865.69811)
{!} 9973
Bruijn
As defined in std/Math.
:import std/Combinator .
:import std/Number .
gcd y [[[=?0 1 (2 0 (1 % 0))]]]
:test ((gcd (+2) (+4)) =? (+2)) ([[1]])
:test ((gcd (+10) (+5)) =? (+5)) ([[1]])
:test ((gcd (+3) (+8)) =? (+1)) ([[1]])C
Iterative Euclid algorithm
int
gcd_iter(int u, int v) {
if (u < 0) u = -u;
if (v < 0) v = -v;
if (v) while ((u %= v) && (v %= u));
return (u + v);
}
Recursive Euclid algorithm
int gcd(int u, int v) {
return (v != 0)?gcd(v, u%v):u;
}
Iterative binary algorithm
int gcd_bin(int u, int v) {
int t, k;
u = u < 0 ? -u : u; /* abs(u) */
v = v < 0 ? -v : v;
if (u < v) {
t = u;
u = v;
v = t;
}
if (v == 0)
return u;
k = 1;
while ((u & 1) == 0 && (v & 1) == 0) { /* u, v - even */
u >>= 1; v >>= 1;
k <<= 1;
}
t = (u & 1) ? -v : u;
while (t) {
while ((t & 1) == 0)
t >>= 1;
if (t > 0)
u = t;
else
v = -t;
t = u - v;
}
return u * k;
}
C#
Iterative
static void Main()
{
Console.WriteLine("GCD of {0} and {1} is {2}", 1, 1, gcd(1, 1));
Console.WriteLine("GCD of {0} and {1} is {2}", 1, 10, gcd(1, 10));
Console.WriteLine("GCD of {0} and {1} is {2}", 10, 100, gcd(10, 100));
Console.WriteLine("GCD of {0} and {1} is {2}", 5, 50, gcd(5, 50));
Console.WriteLine("GCD of {0} and {1} is {2}", 8, 24, gcd(8, 24));
Console.WriteLine("GCD of {0} and {1} is {2}", 36, 17, gcd(36, 17));
Console.WriteLine("GCD of {0} and {1} is {2}", 36, 18, gcd(36, 18));
Console.WriteLine("GCD of {0} and {1} is {2}", 36, 19, gcd(36, 19));
for (int x = 1; x < 36; x++)
{
Console.WriteLine("GCD of {0} and {1} is {2}", 36, x, gcd(36, x));
}
Console.Read();
}
/// <summary>
/// Greatest Common Denominator using Euclidian Algorithm
/// </summary>
static int gcd(int a, int b)
{
while (b != 0) b = a % (a = b);
return a;
}
Example output:
GCD of 1 and 1 is 1 GCD of 1 and 10 is 1 GCD of 10 and 100 is 10 GCD of 5 and 50 is 5 GCD of 8 and 24 is 8 GCD of 36 and 1 is 1 GCD of 36 and 2 is 2 .. GCD of 36 and 16 is 4 GCD of 36 and 17 is 1 GCD of 36 and 18 is 18 .. .. GCD of 36 and 33 is 3 GCD of 36 and 34 is 2 GCD of 36 and 35 is 1
Recursive
static void Main(string[] args)
{
Console.WriteLine("GCD of {0} and {1} is {2}", 1, 1, gcd(1, 1));
Console.WriteLine("GCD of {0} and {1} is {2}", 1, 10, gcd(1, 10));
Console.WriteLine("GCD of {0} and {1} is {2}", 10, 100, gcd(10, 100));
Console.WriteLine("GCD of {0} and {1} is {2}", 5, 50, gcd(5, 50));
Console.WriteLine("GCD of {0} and {1} is {2}", 8, 24, gcd(8, 24));
Console.WriteLine("GCD of {0} and {1} is {2}", 36, 17, gcd(36, 17));
Console.WriteLine("GCD of {0} and {1} is {2}", 36, 18, gcd(36, 18));
Console.WriteLine("GCD of {0} and {1} is {2}", 36, 19, gcd(36, 19));
for (int x = 1; x < 36; x++)
{
Console.WriteLine("GCD of {0} and {1} is {2}", 36, x, gcd(36, x));
}
Console.Read();
}
// Greatest Common Denominator using Euclidian Algorithm
// Gist: https://gist.github.com/SecretDeveloper/6c426f8993873f1a05f7
static int gcd(int a, int b)
{
return b==0 ? a : gcd(b, a % b);
}
Example output:
GCD of 1 and 1 is 1 GCD of 1 and 10 is 1 GCD of 10 and 100 is 10 GCD of 5 and 50 is 5 GCD of 8 and 24 is 8 GCD of 36 and 1 is 1 GCD of 36 and 2 is 2 .. GCD of 36 and 16 is 4 GCD of 36 and 17 is 1 GCD of 36 and 18 is 18 .. .. GCD of 36 and 33 is 3 GCD of 36 and 34 is 2 GCD of 36 and 35 is 1
C++
#include <iostream>
#include <numeric>
int main() {
std::cout << "The greatest common divisor of 12 and 18 is " << std::gcd(12, 18) << " !\n";
}
#include <boost/math/common_factor.hpp>
#include <iostream>
int main() {
std::cout << "The greatest common divisor of 12 and 18 is " << boost::math::gcd(12, 18) << "!\n";
}
- Output:
The greatest common divisor of 12 and 18 is 6!
Clojure
Euclid's Algorithm
(defn gcd
"(gcd a b) computes the greatest common divisor of a and b."
[a b]
(if (zero? b)
a
(recur b (mod a b))))
That recur call is the same as (gcd b (mod a b)), but makes use of Clojure's explicit tail call optimization.
This can be easily extended to work with variadic arguments:
(defn gcd*
"greatest common divisor of a list of numbers"
[& lst]
(reduce gcd
lst))
Stein's Algorithm (Binary GCD)
(defn stein-gcd [a b]
(cond
(zero? a) b
(zero? b) a
(and (even? a) (even? b)) (* 2 (stein-gcd (unsigned-bit-shift-right a 1) (unsigned-bit-shift-right b 1)))
(and (even? a) (odd? b)) (recur (unsigned-bit-shift-right a 1) b)
(and (odd? a) (even? b)) (recur a (unsigned-bit-shift-right b 1))
(and (odd? a) (odd? b)) (recur (unsigned-bit-shift-right (Math/abs (- a b)) 1) (min a b))))
CLU
gcd = proc (a, b: int) returns (int)
while b~=0 do
a, b := b, a//b
end
return(a)
end gcd
start_up = proc()
po: stream := stream$primary_input()
as: array[int] := array[int]$[18, 1071, 3528]
bs: array[int] := array[int]$[12, 1029, 3780]
for i: int in array[int]$indexes(as) do
stream$putl(po, "gcd(" || int$unparse(as[i]) || ", "
|| int$unparse(bs[i]) || ") = "
|| int$unparse(gcd(as[i], bs[i])))
end
end start_up- Output:
gcd(18, 12) = 6 gcd(1071, 1029) = 21 gcd(3528, 3780) = 252
COBOL
IDENTIFICATION DIVISION.
PROGRAM-ID. GCD.
DATA DIVISION.
WORKING-STORAGE SECTION.
01 A PIC 9(10) VALUE ZEROES.
01 B PIC 9(10) VALUE ZEROES.
01 TEMP PIC 9(10) VALUE ZEROES.
PROCEDURE DIVISION.
Begin.
DISPLAY "Enter first number, max 10 digits."
ACCEPT A
DISPLAY "Enter second number, max 10 digits."
ACCEPT B
IF A < B
MOVE B TO TEMP
MOVE A TO B
MOVE TEMP TO B
END-IF
PERFORM UNTIL B = 0
MOVE A TO TEMP
MOVE B TO A
DIVIDE TEMP BY B GIVING TEMP REMAINDER B
END-PERFORM
DISPLAY "The gcd is " A
STOP RUN.
Cobra
class Rosetta
def gcd(u as number, v as number) as number
u, v = u.abs, v.abs
while v > 0
u, v = v, u % v
return u
def main
print "gcd of [12] and [8] is [.gcd(12, 8)]"
print "gcd of [12] and [-8] is [.gcd(12, -8)]"
print "gcd of [96] and [27] is [.gcd(27, 96)]"
print "gcd of [51] and [34] is [.gcd(34, 51)]"Output:
gcd of 12 and 8 is 4 gcd of 12 and -8 is 4 gcd of 96 and 27 is 3 gcd of 51 and 34 is 17
CoffeeScript
Simple recursion
gcd = (x, y) ->
if y == 0 then x else gcd y, x % y
Since JS has no TCO, here's a version with no recursion
gcd = (x, y) ->
[1..(Math.min x, y)].reduce (acc, v) ->
if x % v == 0 and y % v == 0 then v else acc
Common Lisp
Common Lisp provides a gcd function.
CL-USER> (gcd 2345 5432)
7
Here is an implementation using the do macro. We call the function gcd* so as not to conflict with common-lisp:gcd.
(defun gcd* (a b)
(do () ((zerop b) (abs a))
(shiftf a b (mod a b))))
Here is a tail-recursive implementation.
(defun gcd* (a b)
(if (zerop b)
a
(gcd2 b (mod a b))))
The last implementation is based on the loop macro.
(defun gcd* (a b)
(loop for x = a then y
and y = b then (mod x y)
until (zerop y)
finally (return x)))
Component Pascal
BlackBox Component Builder
MODULE Operations;
IMPORT StdLog,Args,Strings;
PROCEDURE Gcd(a,b: LONGINT):LONGINT;
VAR
r: LONGINT;
BEGIN
LOOP
r := a MOD b;
IF r = 0 THEN RETURN b END;
a := b;b := r
END
END Gcd;
PROCEDURE DoGcd*;
VAR
x,y,done: INTEGER;
p: Args.Params;
BEGIN
Args.Get(p);
IF p.argc >= 2 THEN
Strings.StringToInt(p.args[0],x,done);
Strings.StringToInt(p.args[1],y,done);
StdLog.String("gcd("+p.args[0]+","+p.args[1]+")=");StdLog.Int(Gcd(x,y));StdLog.Ln
END
END DoGcd;
END Operations.
Execute:
^Q Operations.DoGcd 12 8 ~
^Q Operations.DoGcd 100 5 ~
^Q Operations.DoGcd 7 23 ~
^Q Operations.DoGcd 24 -112 ~
Output:
gcd(12 ,8 )= 4 gcd(100 ,5 )= 5 gcd(7 ,23 )= 1 gcd(24 ,-112 )= -8
D
import std.stdio, std.numeric;
long myGCD(in long x, in long y) pure nothrow @nogc {
if (y == 0)
return x;
return myGCD(y, x % y);
}
void main() {
gcd(15, 10).writeln; // From Phobos.
myGCD(15, 10).writeln;
}
- Output:
5 5
Dc
[dSa%Lard0<G]dsGx+This code assumes that there are two integers on the stack.
dc -e'28 24 [dSa%Lard0<G]dsGx+ p'
Delphi
See #Pascal / Delphi / Free Pascal.
Draco
proc nonrec gcd(word m, n) word:
word t;
while n ~= 0 do
t := m;
m := n;
n := t % n
od;
m
corp
proc nonrec show(word m, n) void:
writeln("gcd(", m, ", ", n, ") = ", gcd(m, n))
corp
proc nonrec main() void:
show(18, 12);
show(1071, 1029);
show(3528, 3780)
corp- Output:
gcd(18, 12) = 6 gcd(1071, 1029) = 21 gcd(3528, 3780) = 252
dt
[[a b]: a [b a b % gcd] b do?] \gcd defDuckDB
gcd()
DuckDB has a scalar function, gcd(), which is an alias for greatest_common_divisor(). In the following typescript, "D " is the DuckDB prompt.
D select gcd(12345, 9876); ┌──────────────────┐ │ gcd(12345, 9876) │ │ int64 │ ├──────────────────┤ │ 2469 │ └──────────────────┘
Computing the GCD
create or replace function mygcd(a0, b0) as (
with recursive cte as (
select a0 as a, b0 as b
union all
select b as a, mod(a,b) as b
from cte
where b != 0
)
select last(a)
from cte
where b=0
);
Example:
D select mygcd(12345, 9876) as gcd; ┌───────┐ │ gcd │ │ int32 │ ├───────┤ │ 2469 │ └───────┘
DWScript
PrintLn(Gcd(231, 210));
Output:
21
Dyalect
func gcd(a, b) {
func bgcd(a, b, res) {
if a == b {
return res * a
} else if a % 2 == 0 && b % 2 == 0 {
return bgcd(a/2, b/2, 2*res)
} else if a % 2 == 0 {
return bgcd(a/2, b, res)
} else if b % 2 == 0 {
return bgcd(a, b/2, res)
} else if a > b {
return bgcd(a-b, b, res)
} else {
return bgcd(a, b-a, res)
}
}
return bgcd(a, b, 1)
}
var testdata = [
(a: 33, b: 77),
(a: 49865, b: 69811)
]
for v in testdata {
print("gcd(\(v.a), \(v.b)) = \(gcd(v.a, v.b))")
}- Output:
gcd(33, 77) = 11 gcd(49865, 69811) = 9973
E
def gcd(var u :int, var v :int) {
while (v != 0) {
def r := u %% v
u := v
v := r
}
return u.abs()
}EasyLang
func gcd a b .
while b <> 0
swap a b
b = b mod a
.
return a
.
print gcd 120 35
EDSAC order code
The EDSAC had no division instruction, so the GCD routine below has to include its own code for division.
[Greatest common divisor for Rosetta Code.
Program for EDSAC, Initial Orders 2.]
[Library subroutine R2. Reads positive integers during input of orders,
and is then overwritten (so doesn't take up any memory).
Negative numbers can be input by adding 2^35.
Each integer is followed by 'F', except the last is followed by '#TZ'.]
T 45 K [store address in location 45
values are then accessed by code letter H]
P 220 F [<------ address here]
GKT20FVDL8FA40DUDTFI40FA40FS39FG@S2FG23FA5@T5@E4@E13Z
T #H [Tell R2 the storage location defined above]
[Integers to be read by R2. First item is count, then pairs for GCD algo.]
4F 1066F 2019F 1815F 1914F 103785682F 167928761F 109876463F 177777648#TZ
[----------------------------------------------------------------------
Library subroutine P7.
Prints long strictly positive integer at 0D.
10 characters, right justified, padded left with spaces.
Closed, even; 35 storage locations; working position 4D.]
T 56 K
GKA3FT26@H28#@NDYFLDT4DS27@TFH8@S8@T1FV4DAFG31@SFLDUFOFFFSFL4F
T4DA1FA27@G11@XFT28#ZPFT27ZP1024FP610D@524D!FO30@SFL8FE22@
[---------------------------------------------------------------
Subroutine to return GCD of two non-negative 35-bit integers.
Input: Integers at 4D, 6D.
Output: GCD at 4D; changes 6D.
41 locations; working location 0D.]
T 100 K
G K
A 3 F [plant link]
T 39 @
S 4 D [test for 4D = 0]
E 37 @ [if so, quick exit, GCD = 6D]
T 40 @ [clear acc]
[5] A 4 D [load divisor]
[6] T D [initialize shifted divisor]
A 6 D [load dividend]
R D [shift 1 right]
S D [shifted divisor > dividend/2 yet?]
G 15 @ [yes, start subtraction]
T 40 @ [no, clear acc]
A D [shift divisor 1 more]
L D
E 6 @ [loop back (always, since acc = 0)]
[15] T 40 @ [clear acc]
[16] A 6 D [load remainder (initially = dividend)]
S D [trial subtraction]
G 20 @ [skip if can't subtract]
T 6 D [update remainder]
[20] T 40 @ [clear acc]
A 4 D [load original divisor]
S D [is shifted divisor back to original?]
E 29 @ [yes, jump out with acc = 0]
T 40 @ [no, clear acc]
A D [shift divisor 1 right]
R D
T D
E 16 @ [loop back (always, since acc = 0)]
[Here when finished dividing 6D by 4D.
Remainder is at 6D; acc = 0.]
[29] S 6 D [test for 6D = 0]
E 39 @ [if so, exit with GCD in 4D]
T D [else swap 4D and 6D]
A 4 D
T 6 D
S D
T 4 D
E 5 @ [loop back]
[37] A 6 D [here if 4D = 0 at start; GCD is 6D]
T 4 D [return in 4D as GCD]
[39] E F
[40] P F [junk word, to clear accumulator]
[----------------------------------------------------------------------
Main routine]
T 150 K
G K
[Variable]
[0] P F
[Constants]
[1] P D [single-word 1]
[2] A 2#H [order to load first number of first pair]
[3] P 2 F [to inc addresses by 2]
[4] # F [figure shift]
[5] K 2048 F [letter shift]
[6] G F [letters to print 'GCD']
[7] C F
[8] D F
[9] V F [equals sign (in figures mode)]
[10] ! F [space]
[11] @ F [carriage return]
[12] & F [line feed]
[13] K 4096 F [null char]
[Enter here with acc = 0]
[14] O 4 @ [set teleprinter to figures]
S H [negative of number of pairs]
T @ [initialize counter]
A 2 @ [initial load order]
[18] U 23 @ [plant order to load 1st integer]
U 32 @
A 3 @ [inc address by 2]
U 28 @ [plant order to load 2nd integer]
T 34 @
[23] A #H [load 1st integer (order set up at runtime)]
T D [to 0D for printing]
A 25 @ [for return from print subroutine]
G 56 F [print 1st number]
O 10 @ [followed by space]
[28] A #H [load 2nd integer (order set up at runtime)]
T D [to 0D for printing]
A 30 @ [for return from print subroutine]
G 56 F [print 2nd number]
[32] A #H [load 1st integer (order set up at runtime)]
T 4 D [to 4D for GCD subroutine]
[34] A #H [load 2nd integer (order set up at runtime)]
T 6 D [to 6D for GCD subroutine]
[36] A 36 @ [for return from subroutine]
G 100 F [call subroutine for GCD]
[Cosmetic printing, add ' GCD = ']
O 10 @
O 10 @
O 5 @
O 6 @
O 7 @
O 8 @
O 4 @
O 10 @
O 9 @
O 10 @
A 4 D [load GCD]
T D [to 0D for printing]
A 50 @ [for return from print subroutine]
G 56 F [print GCD]
O 11 @ [followed by new line]
O 12 @
[On to next pair]
A @ [load negative count of c.f.s]
A 1 @ [add 1]
E 62 @ [exit if count = 0]
T @ [store back]
A 23 @ [order to load first of pair]
A 3 @ [inc address by 4 for next c.f.]
A 3 @
G 18 @ [loop back (always, since 'A' < 0)]
[62] O 13 @ [null char to flush teleprinter buffer]
Z F [stop]
E 14 Z [define entry point]
P F [acc = 0 on entry]- Output:
1066 2019 GCD = 1
1815 1914 GCD = 33
103785682 167928761 GCD = 1001
109876463 177777648 GCD = 1234567
Eiffel
class
APPLICATION
create
make
feature -- Implementation
gcd (x: INTEGER y: INTEGER): INTEGER
do
if y = 0 then
Result := x
else
Result := gcd (y, x \\ y);
end
end
feature {NONE} -- Initialization
make
-- Run application.
do
print (gcd (15, 10))
print ("%N")
end
end
Elena
ELENA 4.x :
import system'math;
import extensions;
gcd(a,b)
{
var i := a;
var j := b;
while(j != 0)
{
var tmp := i;
i := j;
j := tmp.mod(j)
};
^ i
}
printGCD(a,b)
{
console.printLineFormatted("GCD of {0} and {1} is {2}", a, b, gcd(a,b))
}
public program()
{
printGCD(1,1);
printGCD(1,10);
printGCD(10,100);
printGCD(5,50);
printGCD(8,24);
printGCD(36,17);
printGCD(36,18);
printGCD(36,19);
printGCD(36,33);
}- Output:
GCD of 1 and 1 is 1 GCD of 1 and 10 is 1 GCD of 10 and 100 is 10 GCD of 5 and 50 is 5 GCD of 8 and 24 is 8 GCD of 36 and 17 is 1 GCD of 36 and 18 is 18 GCD of 36 and 19 is 1 GCD of 36 and 33 is 3
Elixir
defmodule RC do
def gcd(a,0), do: abs(a)
def gcd(a,b), do: gcd(b, rem(a,b))
end
IO.puts RC.gcd(1071, 1029)
IO.puts RC.gcd(3528, 3780)
- Output:
21 252
Elm
import Html exposing (Html, div, h1, p, text)
import Html.Attributes exposing (style)
-- Test cases
nr1 : Int
nr1 =
2 * 3 * 5 * 7 * 11
nr2 : Int
nr2 =
7 * 11 * 13 * 17 * 23
main : Html msg
main =
div [ style "margin" "5%", style "font-size" "1.5em", style "color" "blue" ]
[ h1 [ style "font-size" "1.5em" ] [ text "GCD Calculator" ]
, text
("number a = "
++ String.fromInt nr1
++ ", number b = "
++ String.fromInt nr2
)
, p [] [ text ("GCD = " ++ String.fromInt (gcd nr1 nr2)) ]
]
gcd : Int -> Int -> Int
gcd anr bnr =
if bnr /= 0 then
gcd bnr (modBy bnr anr)
else
abs anr
- Output:
number a = 2310, number b = 391391 GCD = 77
Emacs Lisp
(defun gcd (a b)
(cond
((< a b) (gcd a (- b a)))
((> a b) (gcd (- a b) b))
(t a)))
EMal
fun gcd ← int by int a, int b
while b ≠ 0
int t ← b
b ← a % b
a ← t
end
return abs(a)
end
writeLine(gcd(0, 0))
writeLine(gcd(0, -10))
writeLine(gcd(-6, 9))
writeLine(gcd(40902, 24140))
writeLine(gcd(9360, 12240))
writeLine(gcd(123456789, 987654321))- Output:
0 10 3 34 720 9
Erlang
% Implemented by Arjun Sunel
-module(gcd).
-export([main/0]).
main() ->gcd(-36,4).
gcd(A, 0) -> A;
gcd(A, B) -> gcd(B, A rem B).
- Output:
4
ERRE
This is a iterative version.
PROGRAM EUCLIDE
! calculate G.C.D. between two integer numbers
! using Euclidean algorithm
!VAR J%,K%,MCD%,A%,B%
BEGIN
PRINT(CHR$(12);"Input two numbers : ";) !CHR$(147) in C-64 version
INPUT(J%,K%)
A%=J% B%=K%
WHILE A%<>B% DO
IF A%>B%
THEN
A%=A%-B%
ELSE
B%=B%-A%
END IF
END WHILE
MCD%=A%
PRINT("G.C.D. between";J%;"and";K%;"is";MCD%)
END PROGRAM- Output:
Input two numbers : ? 112,44 G.C.D. between 112 and 44 is 4
Euler
The original Euler didn't have built-in loops, this defines a while-loop procedure, as in the Steady Squares#Euler sample.
begin
new while; new gcd;
while <- ` formal condition; formal loopBody;
begin
label again;
again: if condition then begin loopBody; goto again end else 0
end
'
;
gcd <- ` formal m; formal n;
begin
new a; new b; new newA;
a <- abs m;
b <- abs n;
while( ` b <> 0 '
, ` begin
newA <- b;
b <- a mod b;
a <- newA
end
'
);
a
end
'
;
out gcd( -21, 35 )
end $
Euler Math Toolbox
Non-recursive version in Euler Math Toolbox. Note, that there is a built-in command.
>ggt(123456795,1234567851)
33
>function myggt (n:index, m:index) ...
$ if n<m then {n,m}={m,n}; endif;
$ repeat
$ k=mod(n,m);
$ if k==0 then return m; endif;
$ if k==1 then return 1; endif;
$ {n,m}={m,k};
$ end;
$ endfunction
>myggt(123456795,1234567851)
33
Euphoria
Iterative Euclid algorithm
function gcd_iter(integer u, integer v)
integer t
while v do
t = u
u = v
v = remainder(t, v)
end while
if u < 0 then
return -u
else
return u
end if
end functionRecursive Euclid algorithm
function gcd(integer u, integer v)
if v then
return gcd(v, remainder(u, v))
elsif u < 0 then
return -u
else
return u
end if
end functionIterative binary algorithm
function gcd_bin(integer u, integer v)
integer t, k
if u < 0 then -- abs(u)
u = -u
end if
if v < 0 then -- abs(v)
v = -v
end if
if u < v then
t = u
u = v
v = t
end if
if v = 0 then
return u
end if
k = 1
while and_bits(u,1) = 0 and and_bits(v,1) = 0 do
u = floor(u/2) -- u >>= 1
v = floor(v/2) -- v >>= 1
k *= 2 -- k <<= 1
end while
if and_bits(u,1) then
t = -v
else
t = u
end if
while t do
while and_bits(t, 1) = 0 do
t = floor(t/2)
end while
if t > 0 then
u = t
else
v = -t
end if
t = u - v
end while
return u * k
end functionExcel
Excel's GCD can handle multiple values. Type in a cell:
=GCD(A1:E1)- Sample Output:
This will get the GCD of the first 5 cells of the first row.
30 10 500 25 1000 5
Ezhil
## இந்த நிரல் இரு எண்களுக்கு இடையிலான மீச்சிறு பொது மடங்கு (LCM), மீப்பெரு பொது வகுத்தி (GCD) என்ன என்று கணக்கிடும்
நிரல்பாகம் மீபொவ(எண்1, எண்2)
@(எண்1 == எண்2) ஆனால்
## இரு எண்களும் சமம் என்பதால், அந்த எண்ணேதான் அதன் மீபொவ
பின்கொடு எண்1
@(எண்1 > எண்2) இல்லைஆனால்
சிறியது = எண்2
பெரியது = எண்1
இல்லை
சிறியது = எண்1
பெரியது = எண்2
முடி
மீதம் = பெரியது % சிறியது
@(மீதம் == 0) ஆனால்
## பெரிய எண்ணில் சிறிய எண் மீதமின்றி வகுபடுவதால், சிறிய எண்தான் மீப்பெரு பொதுவகுத்தியாக இருக்கமுடியும்
பின்கொடு சிறியது
இல்லை
தொடக்கம் = சிறியது - 1
நிறைவு = 1
@(எண் = தொடக்கம், எண் >= நிறைவு, எண் = எண் - 1) ஆக
மீதம்1 = சிறியது % எண்
மீதம்2 = பெரியது % எண்
## இரு எண்களையும் மீதமின்றி வகுக்கக்கூடிய பெரிய எண்ணைக் கண்டறிகிறோம்
@((மீதம்1 == 0) && (மீதம்2 == 0)) ஆனால்
பின்கொடு எண்
முடி
முடி
முடி
முடி
அ = int(உள்ளீடு("ஓர் எண்ணைத் தாருங்கள் "))
ஆ = int(உள்ளீடு("இன்னோர் எண்ணைத் தாருங்கள் "))
பதிப்பி "நீங்கள் தந்த இரு எண்களின் மீபொவ (மீப்பெரு பொது வகுத்தி, GCD) = ", மீபொவ(அ, ஆ)
F#
let rec gcd a b =
if b = 0
then abs a
else gcd b (a % b)
>gcd 400 600
val it : int = 200
Factor
: gcd ( a b -- c )
[ abs ] [
[ nip ] [ mod ] 2bi gcd
] if-zero ;
FALSE
10 15$ [0=~][$@$@$@\/*-$]#%. { gcd(10,15)=5 }Fantom
class Main
{
static Int gcd (Int a, Int b)
{
a = a.abs
b = b.abs
while (b > 0)
{
t := a
a = b
b = t % b
}
return a
}
public static Void main()
{
echo ("GCD of 51, 34 is: " + gcd(51, 34))
}
}Fermat
GCD(a,b)Forth
: gcd ( a b -- n )
begin dup while tuck mod repeat drop ;
Fortran
Recursive Euclid algorithm
recursive function gcd_rec(u, v) result(gcd)
integer :: gcd
integer, intent(in) :: u, v
if (mod(u, v) /= 0) then
gcd = gcd_rec(v, mod(u, v))
else
gcd = v
end if
end function gcd_rec
Iterative Euclid algorithm
subroutine gcd_iter(value, u, v)
integer, intent(out) :: value
integer, intent(inout) :: u, v
integer :: t
do while( v /= 0 )
t = u
u = v
v = mod(t, v)
enddo
value = abs(u)
end subroutine gcd_iter
A different version, and implemented as function
function gcd(v, t)
integer :: gcd
integer, intent(in) :: v, t
integer :: c, b, a
b = t
a = v
do
c = mod(a, b)
if ( c == 0) exit
a = b
b = c
end do
gcd = b ! abs(b)
end function gcd
Iterative binary algorithm
subroutine gcd_bin(value, u, v)
integer, intent(out) :: value
integer, intent(inout) :: u, v
integer :: k, t
u = abs(u)
v = abs(v)
if( u < v ) then
t = u
u = v
v = t
endif
if( v == 0 ) then
value = u
return
endif
k = 1
do while( (mod(u, 2) == 0).and.(mod(v, 2) == 0) )
u = u / 2
v = v / 2
k = k * 2
enddo
if( (mod(u, 2) == 0) ) then
t = u
else
t = -v
endif
do while( t /= 0 )
do while( (mod(t, 2) == 0) )
t = t / 2
enddo
if( t > 0 ) then
u = t
else
v = -t
endif
t = u - v
enddo
value = u * k
end subroutine gcd_bin
Notes on performance
gcd_iter(40902, 24140) takes us about 2.8 µsec
gcd_bin(40902, 24140) takes us about 2.5 µsec
Iterative binary algorithm in Fortran 2008
Fortran 2008 introduces new intrinsic functions for integer operations that nowadays usually have hardware support, such as TRAILZ to count trailing zeros.
! Stein’s algorithm implemented in Fortran 2008.
! Translated from my implementation for ATS/Postiats.
elemental function gcd (u, v) result (d)
implicit none
integer, intent(in) :: u, v
integer :: d
integer :: x, y
! gcd(x,y) = gcd(u,v), but x and y are non-negative and x <= y.
x = min (abs (u), abs (v))
y = max (abs (u), abs (v))
if (x == 0) then
d = y
else
d = gcd_pos_pos (x, y)
end if
contains
elemental function gcd_pos_pos (u, v) result (d)
integer, intent(in) :: u, v
integer :: d
integer :: n
integer :: x, y
integer :: p, q
! n = the number of common factors of two in u and v.
n = trailz (ior (u, v))
! Remove the common twos from u and v, giving x and y.
x = ishft (u, -n)
y = ishft (v, -n)
! Make both numbers odd. One of the numbers already was odd.
! There is no effect on the value of their gcd.
x = ishft (x, -trailz (x))
y = ishft (y, -trailz (y))
do while (x /= y)
! If x > y then swap x and y, renaming them p
! and q. Thus p <= q, and gcd(p,q) = gcd(x,y).
p = min (x, y)
q = max (x, y)
x = p ! x remains odd.
q = q - p
y = ishft (q, -trailz (q)) ! Make y odd again.
end do
! Put the common twos back in.
d = ishft (x, n)
end function gcd_pos_pos
end function gcd
program test_gcd
implicit none
interface
elemental function gcd (u, v) result (d)
integer, intent(in) :: u, v
integer :: d
end function gcd
end interface
write (*, '("gcd (0, 0) = ", I0)') gcd (0, 0)
write (*, '("gcd (0, 10) = ", I0)') gcd (0, 10)
write (*, '("gcd (-6, -9) = ", I0)') gcd (-6, -9)
write (*, '("gcd (64 * 5, -16 * 3) = ", I0)') gcd (64 * 5, -16 * 3)
write (*, '("gcd (40902, 24140) = ", I0)') gcd (40902, 24140)
write (*, '("gcd (-40902, 24140) = ", I0)') gcd (-40902, 24140)
write (*, '("gcd (40902, -24140) = ", I0)') gcd (40902, -24140)
write (*, '("gcd (-40902, -24140) = ", I0)') gcd (-40902, -24140)
write (*, '("gcd (24140, 40902) = ", I0)') gcd (24140, 40902)
end program test_gcd
- Output:
gcd (0, 0) = 0 gcd (0, 10) = 10 gcd (-6, -9) = 3 gcd (64 * 5, -16 * 3) = 16 gcd (40902, 24140) = 34 gcd (-40902, 24140) = 34 gcd (40902, -24140) = 34 gcd (-40902, -24140) = 34 gcd (24140, 40902) = 34
Free Pascal
See #Pascal / Delphi / Free Pascal.
Frege
module gcd.GCD where
pure native parseInt java.lang.Integer.parseInt :: String -> Int
gcd' a 0 = a
gcd' a b = gcd' b (a `mod` b)
main args = do
(a:b:_) = args
println $ gcd' (parseInt a) (parseInt b)
Frink
Frink has a builtin gcd[x,y] function that returns the GCD of two integers (which can be arbitrarily large.)
println[gcd[12345,98765]]FunL
FunL has pre-defined function gcd in module integers defined as:
def
gcd( 0, 0 ) = error( 'integers.gcd: gcd( 0, 0 ) is undefined' )
gcd( a, b ) =
def
_gcd( a, 0 ) = a
_gcd( a, b ) = _gcd( b, a%b )
_gcd( abs(a), abs(b) )FutureBasic
This is a nearly-trivial 6-line function, so we've dressed it up a bit to show how easily FB builds a full, interactive application. In FB, when you complete your code and hit RUN, the built app can open in as little as 3 seconds.
It also appears in the dock, where you can run it again with a single click.
The running app looks like this—we added a button to generate random examples to process, and an interactive design that instantly responds to each change in an input field.
begin enum 1 // Object tags
_fldA
_ansA
_fldB
_ansB
_rand
end enum
void local fn BuildMacInterface //15-line GUI
window 1, @"Greatest Common Divisor", ( 0, 0, 380, 130 ), NSWindowStyleMaskTitled + NSWindowStyleMaskClosable
textfield _fldA,,, ( 20, 89, 156, 21 )
ControlSetAlignment( _fldA, NSTextAlignmentRight )
ControlSetFormat( _fldA, @"0123456789", Yes, 18, 0 )
textfield _fldB,,, ( 20, 57, 156, 24 )
ControlSetAlignment( _fldB, NSTextAlignmentRight )
ControlSetFormat( _fldB, @"0123456789", Yes, 18, 0 )
textlabel _ansA, @"= ", ( 182, 91, 185, 16 )
textlabel _ansB, @"= ", ( 182, 62, 185, 16 )
button _rand,,,@"Random demo", ( 129, 13, 122, 32 )
menu 1,,, @"File" : menu 1,0,, @"Close", @"w" : MenuItemSetAction(1,0,@"performClose:")
editmenu 2
WindowMakeFirstResponder( 1, _fldA )
end fn
local fn GCD( a as long, b as long ) as long //the requested function
while b
long c = a mod b
a = b : b = c
wend
end fn = a
void local fn DoDialog( ev as Long, tag as long ) //This makes it interactive
long a, b, c
select ev
case _textFieldDidchange //Find GCD of edit fields' contents
a = fn ControlIntegerValue( _fldA )
b = fn ControlIntegerValue( _fldB )
if a + b == 0 then textlabel _ansA, @"= 0" : textlabel _ansB, @"= 0" : exit fn
c = fn GCD( a, b )
textlabel _ansA, fn stringwithformat(@"= %ld x %ld", c, a / c )
textlabel _ansB, fn stringwithformat(@"= %ld x %ld", c, b / c )
case _btnclick //Fill edit fields with random content, then process
select tag
case _rand
c = rnd(65536)
textfield _fldA,,str( c * rnd(65536) )
textfield _fldB,,str( c * rnd(65536) )
fn DoDialog( _textFieldDidchange, 0 )
end select
case _windowWillClose : end
end select
end fn
fn BuildMacInterface
on dialog fn doDialog
handleevents- Output:
GAP
# Built-in
GcdInt(35, 42);
# 7
# Euclidean algorithm
GcdInteger := function(a, b)
local c;
a := AbsInt(a);
b := AbsInt(b);
while b > 0 do
c := a;
a := b;
b := RemInt(c, b);
od;
return a;
end;
GcdInteger(35, 42);
# 7
Genyris
Recursive
def gcd (u v)
u = (abs u)
v = (abs v)
cond
(equal? v 0) u
else (gcd v (% u v))Iterative
def gcd (u v)
u = (abs u)
v = (abs v)
while (not (equal? v 0))
var tmp (% u v)
u = v
v = tmp
uGML
var n,m,r;
n = max(argument0,argument1);
m = min(argument0,argument1);
while (m != 0)
{
r = n mod m;
n = m;
m = r;
}
return a;gnuplot
gcd (a, b) = b == 0 ? a : gcd (b, a % b)
Example:
print gcd (111111, 1111)
Output:
11
Go
Binary Euclidian
package main
import "fmt"
func gcd(a, b int) int {
var bgcd func(a, b, res int) int
bgcd = func(a, b, res int) int {
switch {
case a == b:
return res * a
case a % 2 == 0 && b % 2 == 0:
return bgcd(a/2, b/2, 2*res)
case a % 2 == 0:
return bgcd(a/2, b, res)
case b % 2 == 0:
return bgcd(a, b/2, res)
case a > b:
return bgcd(a-b, b, res)
default:
return bgcd(a, b-a, res)
}
}
return bgcd(a, b, 1)
}
func main() {
type pair struct {
a int
b int
}
var testdata []pair = []pair{
pair{33, 77},
pair{49865, 69811},
}
for _, v := range testdata {
fmt.Printf("gcd(%d, %d) = %d\n", v.a, v.b, gcd(v.a, v.b))
}
}
- Output for Binary Euclidian algorithm:
gcd(33, 77) = 11 gcd(49865, 69811) = 9973
Iterative
package main
import "fmt"
func gcd(x, y int) int {
for y != 0 {
x, y = y, x%y
}
return x
}
func main() {
fmt.Println(gcd(33, 77))
fmt.Println(gcd(49865, 69811))
}
Builtin
(This is just a wrapper for big.GCD)
package main
import (
"fmt"
"math/big"
)
func gcd(x, y int64) int64 {
return new(big.Int).GCD(nil, nil, big.NewInt(x), big.NewInt(y)).Int64()
}
func main() {
fmt.Println(gcd(33, 77))
fmt.Println(gcd(49865, 69811))
}
- Output in either case:
11 9973
Golfscript
;'2706 410'
~{.@\%.}do;- Output:
82
Groovy
Recursive
def gcdR
gcdR = { m, n -> m = m.abs(); n = n.abs(); n == 0 ? m : m%n == 0 ? n : gcdR(n, m%n) }
Iterative
def gcdI = { m, n -> m = m.abs(); n = n.abs(); n == 0 ? m : { while(m%n != 0) { t=n; n=m%n; m=t }; n }() }
Test program:
println " R I"
println "gcd(28, 0) = ${gcdR(28, 0)} == ${gcdI(28, 0)}"
println "gcd(0, 28) = ${gcdR(0, 28)} == ${gcdI(0, 28)}"
println "gcd(0, -28) = ${gcdR(0, -28)} == ${gcdI(0, -28)}"
println "gcd(70, -28) = ${gcdR(70, -28)} == ${gcdI(70, -28)}"
println "gcd(70, 28) = ${gcdR(70, 28)} == ${gcdI(70, 28)}"
println "gcd(28, 70) = ${gcdR(28, 70)} == ${gcdI(28, 70)}"
println "gcd(800, 70) = ${gcdR(800, 70)} == ${gcdI(800, 70)}"
println "gcd(27, -70) = ${gcdR(27, -70)} == ${gcdI(27, -70)}"
Output:
R I gcd(28, 0) = 28 == 28 gcd(0, 28) = 28 == 28 gcd(0, -28) = 28 == 28 gcd(70, -28) = 14 == 14 gcd(70, 28) = 14 == 14 gcd(28, 70) = 14 == 14 gcd(800, 70) = 10 == 10 gcd(27, -70) = 1 == 1
Haskell
That is already available as the function gcd in the Prelude. Here's the implementation, with one name adjusted to avoid a Wiki formatting glitch:
gcd :: (Integral a) => a -> a -> a
gcd x y = gcd_ (abs x) (abs y)
where
gcd_ a 0 = a
gcd_ a b = gcd_ b (a `rem` b)
HicEst
FUNCTION gcd(a, b)
IF(b == 0) THEN
gcd = ABS(a)
ELSE
aa = a
gcd = b
DO i = 1, 1E100
r = ABS(MOD(aa, gcd))
IF( r == 0 ) RETURN
aa = gcd
gcd = r
ENDDO
ENDIF
END
Hoon
::
:: Greatest common divisor (gcd), Euclid's algorithm.
::
|= [a=@ud b=@ud]
^- @ud
?> (gth b 0)
?: =((mod a b) 0)
b
$(a b, b (mod a b))An example of use at dojo (assuming the gate is pinned as gcd): > (gcd 123 36) 3
Icon and Unicon
numbers implements this as:
J
x+.y
For example:
12 +. 30
6
Note that +. is a single, two character token. GCD is a primitive in J (and anyone that has studied the right kind of mathematics should instantly recognize why the same operation is used for both GCD and OR -- among other things, GCD and boolean OR both have the same identity element: 0, and of course they produce the same numeric results on the same arguments (when we are allowed to use the usual 1 bit implementation of 0 and 1 for false and true) - more than that, though, GCD corresponds to George Boole's original "Boolean Algebra" (as it was later called). The redefinition of "Boolean algebra" to include logical negation came much later, in the 20th century).
gcd could also be defined recursively, if you do not mind a little inefficiency:
gcd=: (| $: [)^:(0<[)&|
Java
From javax.swing.table.DefaultTableModel
/* recursive */
int gcd(int a, int b) {
return (b == 0) ? a : gcd(b, a % b);
}
Iterative
public static long gcd(long a, long b){
long factor= Math.min(a, b);
for(long loop= factor;loop > 1;loop--){
if(a % loop == 0 && b % loop == 0){
return loop;
}
}
return 1;
}
Iterative Euclid's Algorithm
public static int gcd(int a, int b) //valid for positive integers.
{
while(b > 0)
{
int c = a % b;
a = b;
b = c;
}
return a;
}
Optimized Iterative
static int gcd(int a,int b)
{
int min=a>b?b:a,max=a+b-min, div=min;
for(int i=1;i<min;div=min/++i)
if(min%div==0&&max%div==0)
return div;
return 1;
}
Iterative binary algorithm
public static long gcd(long u, long v){
long t, k;
if (v == 0) return u;
u = Math.abs(u);
v = Math.abs(v);
if (u < v){
t = u;
u = v;
v = t;
}
for(k = 1; (u & 1) == 0 && (v & 1) == 0; k <<= 1){
u >>= 1; v >>= 1;
}
t = (u & 1) != 0 ? -v : u;
while (t != 0){
while ((t & 1) == 0) t >>= 1;
if (t > 0)
u = t;
else
v = -t;
t = u - v;
}
return u * k;
}Recursive
public static long gcd(long a, long b){
if(a == 0) return b;
if(b == 0) return a;
if(a > b) return gcd(b, a % b);
return gcd(a, b % a);
}Built-in
import java.math.BigInteger;
public static long gcd(long a, long b){
return BigInteger.valueOf(a).gcd(BigInteger.valueOf(b)).longValue();
}JavaScript
Iterative implementation
function gcd(a,b) {
a = Math.abs(a);
b = Math.abs(b);
if (b > a) {
var temp = a;
a = b;
b = temp;
}
while (true) {
a %= b;
if (a === 0) { return b; }
b %= a;
if (b === 0) { return a; }
}
}Recursive.
function gcd_rec(a, b) {
return b ? gcd_rec(b, a % b) : Math.abs(a);
}Implementation that works on an array of integers.
function GCD(arr) {
var i, y,
n = arr.length,
x = Math.abs(arr[0]);
for (i = 1; i < n; i++) {
y = Math.abs(arr[i]);
while (x && y) {
(x > y) ? x %= y : y %= x;
}
x += y;
}
return x;
}
//For example:
GCD([57,0,-45,-18,90,447]); //=> 3Joy
DEFINE gcd == [0 >] [dup rollup rem] while pop.jq
def recursive_gcd(a; b):
if b == 0 then a
else recursive_gcd(b; a % b)
end ;Recent versions of jq include support for tail recursion optimization for arity-0 filters (which can be thought of as arity-1 functions), so here is an implementation that takes advantage of that optimization. Notice that the subfunction, rgcd, can be easily derived from recursive_gcd above by moving the arguments to the input:
def gcd(a; b):
# The subfunction expects [a,b] as input
# i.e. a ~ .[0] and b ~ .[1]
def rgcd: if .[1] == 0 then .[0]
else [.[1], .[0] % .[1]] | rgcd
end;
[a,b] | rgcd ;Julia
Julia includes a built-in gcd function:
julia> gcd(4,12) 4 julia> gcd(6,12) 6 julia> gcd(7,12) 1
The actual implementation of this function in Julia 0.2's standard library is reproduced here:
function gcd{T<:Integer}(a::T, b::T)
neg = a < 0
while b != 0
t = b
b = rem(a, b)
a = t
end
g = abs(a)
neg ? -g : g
end(For arbitrary-precision integers, Julia calls a different implementation from the GMP library.)
K
K3
gcd:{:[~x;y;_f[y;x!y]]}K6
gcd:{$[~x;y;o[x!y;x]]}Klong
gcd::{:[~x;y:|~y;x:|x>y;.f(y;x!y);.f(x;y!x)]}Kotlin
Recursive one line solution:
tailrec fun gcd(a: Int, b: Int): Int = if (b == 0) kotlin.math.abs(a) else gcd(b, a % b)LabVIEW
It may be helpful to read about Recursion in LabVIEW.
This image is a VI Snippet, an executable image of LabVIEW code. The LabVIEW version is shown on the top-right hand corner. You can download it, then drag-and-drop it onto the LabVIEW block diagram from a file browser, and it will appear as runnable, editable code.
Lambdatalk
{def gcd
{lambda {:a :b}
{if {= :b 0}
then :a
else {gcd :b {% :a :b}}}}}
-> gcd
{gcd 12 3}
-> 3
{gcd 123 122}
-> 1
{S.map {gcd 123} {S.serie 1 30}}
-> 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3
A simpler one if a and b are greater than zero
{def GCD
{lambda {:a :b}
{if {= :a :b}
then :a
else {if {> :a :b}
then {GCD {- :a :b} :b}
else {GCD :a {- :b :a}}}}}}
-> GCD
{S.map {GCD 123} {S.serie 1 30}}
-> 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3LFE
> (defun gcd
"Get the greatest common divisor."
((a 0) a)
((a b) (gcd b (rem a b))))Usage:
> (gcd 12 8) 4 > (gcd 12 -8) 4 > (gcd 96 27) 3 > (gcd 51 34) 17
Limbo
gcd(x: int, y: int): int
{
if(y == 0)
return x;
return gcd(y, x % y);
}LiveCode
function gcd x,y
repeat until y = 0
put x mod y into z
put y into x
put z into y
end repeat
return x
end gcdLogo
to gcd :a :b
if :b = 0 [output :a]
output gcd :b modulo :a :b
endLOLCODE
HAI 1.3
HOW IZ I gcd YR a AN YR b
a R BIGGR OF a AN PRODUKT OF a AN -1 BTW absolute value of a
b R BIGGR OF b AN PRODUKT OF b AN -1 BTW absolute value of b
BOTH SAEM a AN b, O RLY?
YA RLY
FOUND YR a
OIC
BOTH SAEM a AN 0, O RLY?
YA RLY
FOUND YR b
OIC
BOTH SAEM b AN 0, O RLY?
YA RLY
FOUND YR a
OIC
BOTH SAEM b AN BIGGR OF a AN b, O RLY? BTW make sure a is the larger of (a, b)
YA RLY
I HAS A temp ITZ a
a R b
b R temp
OIC
IM IN YR loop
I HAS A temp ITZ b
b R MOD OF a AN b
a R temp
BOTH SAEM b AN 0, O RLY?
YA RLY
FOUND YR a
OIC
IM OUTTA YR loop
IF U SAY SO
VISIBLE I IZ gcd YR 40902 AN YR 24140 MKAY
KTHXBYELSE
(*
** MÉTHODE D'EUCLIDE POUR TROUVER LE PLUS GRAND DIVISEUR COMMUN D'UN
** NUMÉRATEUR ET D'UN DÉNOMINATEUR.
*)
PROCÉDURE &PGDC(ENTIER U, ENTIER V) : ENTIER LOCAL U, V
ENTIER T
TANT QUE U > 0 FAIRE
SI U< V ALORS
T<-U
U<-V
V<-T
FIN SI
U <- U - V
BOUCLER
RÉSULTAT V
FIN PROCÉDURE
PROCÉDURE &DEMO(ENTIER U, ENTIER V) LOCAL U, V
AFFICHER ['Le PGDC de ',U,'/',U,' est ',U,/] U, V, &PGDC(U,V)
FIN PROCÉDURE
&DEMO(9,12)
&DEMO(6144,8192)
&DEMO(100,5)
&DEMO(7,23)Resultats:
Le PGDC de 9/12 est 3 Le PGDC de 6144/8192 est 2048 Le PGDC de 100/5 est 5 Le PGDC de 7/23 est 1
Lua
function gcd(a,b)
if b ~= 0 then
return gcd(b, a % b)
else
return math.abs(a)
end
end
function demo(a,b)
print("GCD of " .. a .. " and " .. b .. " is " .. gcd(a, b))
end
demo(100, 5)
demo(5, 100)
demo(7, 23)Output:
GCD of 100 and 5 is 5 GCD of 5 and 100 is 5 GCD of 7 and 23 is 1
Faster iterative solution of Euclid:
function gcd(a,b)
while b~=0 do
a,b=b,a%b
end
return math.abs(a)
endLucid
dataflow algorithm
gcd(n,m) where
z = [% n, m %] fby if x > y then [% x - y, y %] else [% x, y - x%] fi;
x = hd(z);
y = hd(tl(z));
gcd(n, m) = (x asa x*y eq 0) fby eod;
endLuck
function gcd(a: int, b: int): int = (
if a==0 then b
else if b==0 then a
else if a>b then gcd(b, a % b)
else gcd(a, b % a)
)M2000 Interpreter
gcd=lambda (u as long, v as long) -> {
=if(v=0&->abs(u), lambda(v, u mod v))
}
gcd_Iterative= lambda (m as long, n as long) -> {
while m {
let old_m = m
m = n mod m
n = old_m
}
=abs(n)
}
Module CheckGCD (f){
Print f(49865, 69811)=9973
Def ExpType$(x)=Type$(x)
Print ExpType$(f(49865, 69811))="Long"
}
CheckGCD gcd
CheckGCD gcd_Iterativem4
This should work in any POSIX-compliant m4. I have tested it with GNU m4, OpenBSD m4, and Heirloom Devtools m4. It is Euler’s algorithm.
divert(-1)
define(`gcd',
`ifelse(eval(`0 <= (' $1 `)'),`0',`gcd(eval(`-(' $1 `)'),eval(`(' $2 `)'))',
eval(`0 <= (' $2 `)'),`0',`gcd(eval(`(' $1 `)'),eval(`-(' $2 `)'))',
eval(`(' $1 `) == 0'),`0',`gcd(eval(`(' $2 `) % (' $1 `)'),eval(`(' $1 `)'))',
eval(`(' $2 `)'))')
divert`'dnl
dnl
gcd(0, 0) = 0
gcd(24140, 40902) = 34
gcd(-24140, -40902) = 34
gcd(-40902, 24140) = 34
gcd(40902, -24140) = 34- Output:
0 = 0 34 = 34 34 = 34 34 = 34 34 = 34
Maple
To compute the greatest common divisor of two integers in Maple, use the procedure igcd.
igcd( a, b )For example,
> igcd( 24, 15 );
3Mathematica / Wolfram Language
GCD[a, b]MATLAB
function [gcdValue] = greatestcommondivisor(integer1, integer2)
gcdValue = gcd(integer1, integer2);Maxima
/* There is a function gcd(a, b) in Maxima, but one can rewrite it */
gcd2(a, b) := block([a: abs(a), b: abs(b)], while b # 0 do [a, b]: [b, mod(a, b)], a)$
/* both will return 2^97 * 3^48 */
gcd(100!, 6^100), factor;
gcd2(100!, 6^100), factor;MAXScript
Iterative Euclid algorithm
fn gcdIter a b =
(
while b > 0 do
(
c = mod a b
a = b
b = c
)
abs a
)Recursive Euclid algorithm
fn gcdRec a b =
(
if b > 0 then gcdRec b (mod a b) else abs a
)Mercury
Recursive Euclid algorithm
:- module gcd.
:- interface.
:- import_module integer.
:- func gcd(integer, integer) = integer.
:- implementation.
:- pragma memo(gcd/2).
gcd(A, B) = (if B = integer(0) then A else gcd(B, A mod B)).An example console program to demonstrate the gcd module:
:- module test_gcd.
:- interface.
:- import_module io.
:- pred main(io::di, io::uo) is det.
:- implementation.
:- import_module char.
:- import_module gcd.
:- import_module integer.
:- import_module list.
:- import_module string.
main(!IO) :-
command_line_arguments(Args, !IO),
filter(is_all_digits, Args, CleanArgs),
Arg1 = list.det_index0(CleanArgs, 0),
Arg2 = list.det_index0(CleanArgs, 1),
A = integer.det_from_string(Arg1),
B = integer.det_from_string(Arg2),
Fmt = integer.to_string,
GCD = gcd(A, B),
io.format("gcd(%s, %s) = %s\n", [s(Fmt(A)), s(Fmt(B)), s(Fmt(GCD))], !IO).Example output:
gcd(70000000000000000000000, 60000000000000000000000000) = 10000000000000000000000min
((dup 0 !=) (swap over mod) while pop abs) ^gcdMINIL
// Greatest common divisor
00 0E GCD: ENT R0
01 1E ENT R1
02 21 Again: R2 = R1
03 10 Loop: R1 = R0
04 02 R0 = R2
05 2D Minus: DEC R2
06 8A JZ Stop
07 1D DEC R1
08 C5 JNZ Minus
09 83 JZ Loop
0A 1D Stop: DEC R1
0B C2 JNZ Again
0C 80 JZ GCD // Display GCD in R0MiniScript
Using an iterative Euclidean algorithm:
gcd = function(a, b)
while b
temp = b
b = a % b
a = temp
end while
return abs(a)
end function
print gcd(18,12)- Output:
6
MiniZinc
function var int: gcd(int:a2,int:b2) =
let {
int:a1 = max(a2,b2);
int:b1 = min(a2,b2);
array[0..a1,0..b1] of var int: gcd;
constraint forall(a in 0..a1)(
forall(b in 0..b1)(
gcd[a,b] ==
if (b == 0) then
a
else
gcd[b, a mod b]
endif
)
)
} in gcd[a1,b1];
var int: gcd1 = gcd(8,12);
solve satisfy;
output [show(gcd1),"\n"];- Output:
6
MIPS Assembly
gcd:
# a0 and a1 are the two integer parameters
# return value is in v0
move $t0, $a0
move $t1, $a1
loop:
beq $t1, $0, done
div $t0, $t1
move $t0, $t1
mfhi $t1
j loop
done:
move $v0, $t0
jr $raМК-61/52
ИПA ИПB / П9 КИП9 ИПA ИПB ПA ИП9 * - ПB x=0 00 ИПA С/П
Enter: n = РA, m = РB (n > m).
ML
mLite
fun gcd (a, 0) = a
| (0, b) = b
| (a, b) where (a < b)
= gcd (a, b rem a)
| (a, b) = gcd (b, a rem b)ML / Standard ML
See also #Standard ML.
fun gcd a 0 = a
| gcd a b = gcd b (a mod b)Modula-2
MODULE ggTkgV;
FROM InOut IMPORT ReadCard, WriteCard, WriteLn, WriteString, WriteBf;
VAR x, y, u, v : CARDINAL;
BEGIN
WriteString ("x = "); WriteBf; ReadCard (x);
WriteString ("y = "); WriteBf; ReadCard (y);
u := x;
v := y;
WHILE x # y DO
(* ggT (x, y) = ggT (x0, y0), x * v + y * u = 2 * x0 * y0 *)
IF x > y THEN
x := x - y;
u := u + v
ELSE
y := y - x;
v := v + u
END
END;
WriteString ("ggT ="); WriteCard (x, 6); WriteLn;
WriteString ("kgV ="); WriteCard ((u+v) DIV 2, 6); WriteLn;
WriteString ("u ="); WriteCard (u, 6); WriteLn;
WriteString ("v ="); WriteCard (v , 6); WriteLn
END ggTkgV.Producing the output
jan@Beryllium:~/modula/Wirth/PIM$ ggtkgv
x = 12
y = 20
ggT = 4
kgV = 60
u = 44
v = 76
jan@Beryllium:~/modula/Wirth/PIM$ ggtkgv
x = 123
y = 255
ggT = 3
kgV = 10455
u = 13773
v = 7137Modula-3
MODULE GCD EXPORTS Main;
IMPORT IO, Fmt;
PROCEDURE GCD(a, b: CARDINAL): CARDINAL =
BEGIN
IF a = 0 THEN
RETURN b;
ELSIF b = 0 THEN
RETURN a;
ELSIF a > b THEN
RETURN GCD(b, a MOD b);
ELSE
RETURN GCD(a, b MOD a);
END;
END GCD;
BEGIN
IO.Put("GCD of 100, 5 is " & Fmt.Int(GCD(100, 5)) & "\n");
IO.Put("GCD of 5, 100 is " & Fmt.Int(GCD(5, 100)) & "\n");
IO.Put("GCD of 7, 23 is " & Fmt.Int(GCD(7, 23)) & "\n");
END GCD.Output:
GCD of 100, 5 is 5 GCD of 5, 100 is 5 GCD of 7, 23 is 1
Monicelli
#main function (needs two ints from stdin)
Lei ha clacsonato bitumatissimi cari amici ospiti
voglio arrivata, Necchi bitumati qui tra noi benvenuti ENERGIAAh
voglio espertizzata, Necchi
mi porga arrivata mi porga espertizzata bitumata mi raccomando
voglio garantita, Necchi come se fosse prematurata la supercazzola accanita con arrivata, espertizzata o scherziamo?
brematurata la supercazzola novella o scherziamo? bitumata ma dai
garantita a posterdati
# gcd function
blinda la supercazzola Necchi accanita con visibilio Necchi, sgomento Necchi o scherziamo?
voglio la catarsi, Necchi bituma, e come bituma lui non bituma nessuno
voglio l'entusiasmo, Necchi bituma, chi di bituma vive bitumato muore
che cos'è il visibilio? sgomento: vaffanzum visibilio!
o magari maggiore di sgomento: catarsi come fosse visibilio meno sgomento bitumato ma non troppo
entusiasmo come fosse sgomento bitumato anche piu del necessario
o tarapia tapioco: catarsi come fosse sgomento meno il visibilio bitumante dai tempi andati
entusiasmo come fosse visibilio e velocità di esecuzione
voglio la spensierataggine, Necchi come fosse prematurata la supercazzola accanita con catarsi, entusiasmo o scherziamo?
vaffanzum la spensierataggine!
# prints new line
blinda la supercazzola novella o scherziamo?
voglio novita, Mascetti come se fosse 10 bituma come fosse una lungaggine, uno scherzo di mano
novita a posterdati!MUMPS
GCD(A,B)
QUIT:((A/1)'=(A\1))!((B/1)'=(B\1)) 0
SET:A<0 A=-A
SET:B<0 B=-B
IF B'=0
FOR SET T=A#B,A=B,B=T QUIT:B=0 ;ARGUEMENTLESS FOR NEEDS TWO SPACES
QUIT AOuput:
CACHE>S X=$$GCD^ROSETTA(12,24) W X 12 CACHE>S X=$$GCD^ROSETTA(24,-112) W X 8 CACHE>S X=$$GCD^ROSETTA(24,-112.2) W X 0
MySQL
DROP FUNCTION IF EXISTS gcd;
DELIMITER |
CREATE FUNCTION gcd(x INT, y INT)
RETURNS INT
BEGIN
SET @dividend=GREATEST(ABS(x),ABS(y));
SET @divisor=LEAST(ABS(x),ABS(y));
IF @divisor=0 THEN
RETURN @dividend;
END IF;
SET @gcd=NULL;
SELECT gcd INTO @gcd FROM
(SELECT @tmp:=@dividend,
@dividend:=@divisor AS gcd,
@divisor:=@tmp % @divisor AS remainder
FROM mysql.help_relation WHERE @divisor>0) AS x
WHERE remainder=0;
RETURN @gcd;
END;|
DELIMITER ;
SELECT gcd(12345, 9876);+------------------+ | gcd(12345, 9876) | +------------------+ | 2469 | +------------------+ 1 row in set (0.00 sec)
Nanoquery
Iterative
def gcd(a, b)
factor = a.min(b)
for loop in range(factor, 2)
if (a % loop = 0) and (b % loop = 0)
return loop
end
end
return 1
endIterative Euclid's Method
def gcd_euclid(a, b)
while b > 0
c = a % b
a = b
b = c
end
return a
endNetRexx
/* NetRexx */
options replace format comments java crossref symbols nobinary
numeric digits 2000
runSample(arg)
return
-- 09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)~~
-- Euclid's algorithm - iterative implementation
method gcdEucidI(a_, b_) public static
loop while b_ > 0
c_ = a_ // b_
a_ = b_
b_ = c_
end
return a_
-- 09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)~~
-- Euclid's algorithm - recursive implementation
method gcdEucidR(a_, b_) public static
if b_ \= 0 then a_ = gcdEucidR(b_, a_ // b_)
return a_
-- 09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)~~
method runSample(arg) private static
-- pairs of numbers, each number in the pair separated by a colon, each pair separated by a comma
parse arg tests
if tests = '' then
tests = '0:0, 6:4, 7:21, 12:36, 33:77, 41:47, 99:51, 100:5, 7:23, 1989:867, 12345:9876, 40902:24140, 49865:69811, 137438691328:2305843008139952128'
-- most of what follows is for formatting
xiterate = 0
xrecurse = 0
ll_ = 0
lr_ = 0
lgi = 0
lgr = 0
loop i_ = 1 until tests = ''
xiterate[0] = i_
xrecurse[0] = i_
parse tests pair ',' tests
parse pair l_ ':' r_ .
-- get the GCDs
gcdi = gcdEucidI(l_, r_)
gcdr = gcdEucidR(l_, r_)
xiterate[i_] = l_ r_ gcdi
xrecurse[i_] = l_ r_ gcdr
ll_ = ll_.max(l_.strip.length)
lr_ = lr_.max(r_.strip.length)
lgi = lgi.max(gcdi.strip.length)
lgr = lgr.max(gcdr.strip.length)
end i_
-- save formatter sizes in stems
xiterate[-1] = ll_ lr_ lgi
xrecurse[-1] = ll_ lr_ lgr
-- present results
showResults(xiterate, 'Euclid''s algorithm - iterative')
showResults(xrecurse, 'Euclid''s algorithm - recursive')
say
if verifyResults(xiterate, xrecurse) then
say 'Success: Results of iterative and recursive methods match'
else
say 'Error: Results of iterative and recursive methods do not match'
say
return
-- 09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)~~
method showResults(stem, title) public static
say
say title
parse stem[-1] ll lr lg
loop v_ = 1 to stem[0]
parse stem[v_] lv rv gcd .
say lv.right(ll)',' rv.right(lr) ':' gcd.right(lg)
end v_
return
-- 09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)~~
method verifyResults(stem1, stem2) public static returns boolean
if stem1[0] \= stem2[0] then signal BadArgumentException
T = (1 == 1)
F = \T
verified = T
loop i_ = 1 to stem1[0]
if stem1[i_] \= stem2[i_] then do
verified = F
leave i_
end
end i_
return verified- Output:
Euclid's algorithm - iterative
0, 0 : 0
6, 4 : 2
7, 21 : 7
12, 36 : 12
33, 77 : 11
41, 47 : 1
99, 51 : 3
100, 5 : 5
7, 23 : 1
1989, 867 : 51
12345, 9876 : 2469
40902, 24140 : 34
49865, 69811 : 9973
137438691328, 2305843008139952128 : 262144
Euclid's algorithm - recursive
0, 0 : 0
6, 4 : 2
7, 21 : 7
12, 36 : 12
33, 77 : 11
41, 47 : 1
99, 51 : 3
100, 5 : 5
7, 23 : 1
1989, 867 : 51
12345, 9876 : 2469
40902, 24140 : 34
49865, 69811 : 9973
137438691328, 2305843008139952128 : 262144
Success: Results of iterative and recursive methods match
NewLISP
(gcd 12 36)
→ 12Nial
Nial provides gcd in the standard lib.
|loaddefs 'niallib/gcd.ndf'
|gcd 6 4
=2defining it for arrays
# red is the reduction operator for a sorted list
# one is termination condition
red is cull filter (0 unequal) link [mod [rest, first] , first]
one is or [= [1 first, tally], > [2 first, first]]
gcd is fork [one, first, gcd red] sort <=Using it
|gcd 9 6 3
=3Nim
Recursive Euclid algorithm
func gcd_recursive*(u, v: SomeSignedInt): int64 =
if u mod v != 0:
result = gcd_recursive(v, u mod v)
else:
result = abs(v)
when isMainModule:
import strformat
let (x, y) = (49865, 69811)
echo &"gcd({x}, {y}) = {gcd_recursive(49865, 69811)}"- Output:
gcd(49865, 69811) = 9973
Iterative Euclid algorithm
func gcd_iterative*(u, v: SomeSignedInt): int64 =
var u = u
var v = v
while v != 0:
u = u mod v
swap u, v
result = abs(u)
when isMainModule:
import strformat
let (x, y) = (49865, 69811)
echo &"gcd({x}, {y}) = {gcd_iterative(49865, 69811)}")- Output:
gcd(49865, 69811) = 9973
Iterative binary algorithm
template isEven(n: int64): bool = (n and 1) == 0
func gcd_binary*(u, v: int64): int64 =
var u = abs(u)
var v = abs(v)
if u < v: swap u, v
if v == 0: return u
var k = 1
while u.isEven and v.isEven:
u = u shr 1
v = v shr 1
k = k shl 1
var t = if u.isEven: u else: -v
while t != 0:
while t.isEven: t = ashr(t, 1)
if t > 0: u = t
else: v = -t
t = u - v
result = u * k
when isMainModule:
import strformat
let (x, y) = (49865, 69811)
echo &"gcd({x}, {y}) = {gcd_binary(49865, 69811)}"- Output:
gcd(49865, 69811) = 9973
Nu
def 'math gcd' [] {
reduce {|x gcd|
generate { match $in { [$a 0] => {out: $a} [$a $b] => {next: [$b ($a mod $b)]} } } [$x $gcd]
| first
}
}
[63 18 45] | math gcd- Output:
9
Oberon-2
MODULE GCD;
(* Greatest Common Divisor *)
IMPORT
Out;
PROCEDURE Gcd(a,b: LONGINT):LONGINT;
VAR
r: LONGINT;
BEGIN
LOOP
r := a MOD b;
IF r = 0 THEN RETURN b END;
a := b;b := r
END
END Gcd;
BEGIN
Out.String("GCD of 12 and 8 : ");Out.LongInt(Gcd(12,8),4);Out.Ln;
Out.String("GCD of 100 and 5 : ");Out.LongInt(Gcd(100,5),4);Out.Ln;
Out.String("GCD of 7 and 23 : ");Out.LongInt(Gcd(7,23),4);Out.Ln;
Out.String("GCD of 24 and -112 : ");Out.LongInt(Gcd(12,8),4);Out.Ln;
Out.String("GCD of 40902 and 24140 : ");Out.LongInt(Gcd(40902,24140),4);Out.Ln
END GCD.- Output:
GCD of 12 and 8 : 4 GCD of 100 and 5 : 5 GCD of 7 and 23 : 1 GCD of 24 and -112 : 4 GCD of 40902 and 24140 : 34
Objeck
bundle Default {
class GDC {
function : Main(args : String[]), Nil {
for(x := 1; x < 36; x += 1;) {
IO.Console->GetInstance()->Print("GCD of ")->Print(36)->Print(" and ")->Print(x)->Print(" is ")->PrintLine(GDC(36, x));
};
}
function : native : GDC(a : Int, b : Int), Int {
t : Int;
if(a > b) {
t := b; b := a; a := t;
};
while (b <> 0) {
t := a % b; a := b; b := t;
};
return a;
}
}
}OCaml
let rec gcd a = function
| 0 -> a
| b -> gcd b (a mod b)Built-in
#load "nums.cma";;
open Big_int;;
let gcd a b =
int_of_big_int (gcd_big_int (big_int_of_int a) (big_int_of_int b))Octave
r = gcd(a, b)Oforth
gcd is already defined into Integer class :
128 96 gcdSource of this method is (see Integer.of file) :
Integer method: gcd self while ( dup ) [ tuck mod ] drop ;Ol
(print (gcd 1071 1029))
; ==> 21Order
#include <order/interpreter.h>
#define ORDER_PP_DEF_8gcd ORDER_PP_FN( \
8fn(8U, 8V, \
8if(8isnt_0(8V), 8gcd(8V, 8remainder(8U, 8V)), 8U)))
// No support for negative numbersOz
declare
fun {UnsafeGCD A B}
if B == 0 then
A
else
{UnsafeGCD B A mod B}
end
end
fun {GCD A B}
if A == 0 andthen B == 0 then
raise undefined(gcd 0 0) end
else
{UnsafeGCD {Abs A} {Abs B}}
end
end
in
{Show {GCD 456 ~632}}PARI/GP
gcd(a,b)PASCAL program GCF (INPUT, OUTPUT);
var
a,b,c:integer;
begin
writeln('Enter 1st number');
read(a);
writeln('Enter 2nd number');
read(b);
while (a*b<>0)
do
begin
c:=a;
a:=b mod a;
b:=c;
end;
writeln('GCF :=', a+b );
end.
By: NG
Pascal / Delphi / Free Pascal
Recursive Euclid algorithm
PROGRAM EXRECURGCD.PAS;
{$IFDEF FPC}
{$mode objfpc}{$H+}{$J-}{R+}
{$ELSE}
{$APPTYPE CONSOLE}
{$ENDIF}
(*)
Free Pascal Compiler version 3.2.0 [2020/06/14] for x86_64
The free and readable alternative at C/C++ speeds
compiles natively to almost any platform, including raspberry PI
(*)
FUNCTION gcd_recursive(u, v: longint): longint;
BEGIN
IF ( v = 0 ) THEN Exit ( u ) ;
result := gcd_recursive ( v, u MOD v ) ;
END;
BEGIN
WriteLn ( gcd_recursive ( 231, 7 ) ) ;
END.JPD 2021/03/14
Iterative Euclid algorithm
function gcd_iterative(u, v: longint): longint;
var
t: longint;
begin
while v <> 0 do
begin
t := u;
u := v;
v := t mod v;
end;
gcd_iterative := abs(u);
end;Iterative binary algorithm
function gcd_binary(u, v: longint): longint;
var
t, k: longint;
begin
u := abs(u);
v := abs(v);
if u < v then
begin
t := u;
u := v;
v := t;
end;
if v = 0 then
gcd_binary := u
else
begin
k := 1;
while (u mod 2 = 0) and (v mod 2 = 0) do
begin
u := u >> 1;
v := v >> 1;
k := k << 1;
end;
if u mod 2 = 0 then
t := u
else
t := -v;
while t <> 0 do
begin
while t mod 2 = 0 do
t := t div 2;
if t > 0 then
u := t
else
v := -t;
t := u - v;
end;
gcd_binary := u * k;
end;
end;Demo program:
Program GreatestCommonDivisorDemo(output);
begin
writeln ('GCD(', 49865, ', ', 69811, '): ', gcd_iterative(49865, 69811), ' (iterative)');
writeln ('GCD(', 49865, ', ', 69811, '): ', gcd_recursive(49865, 69811), ' (recursive)');
writeln ('GCD(', 49865, ', ', 69811, '): ', gcd_binary (49865, 69811), ' (binary)');
end.Output:
GCD(49865, 69811): 9973 (iterative) GCD(49865, 69811): 9973 (recursive) GCD(49865, 69811): 9973 (binary)
PascalABC.NET
function GCD(a,b: integer): integer;
begin
while b > 0 do
(a,b) := (b,a mod b);
Result := a;
end;
function GCDRec(a,b: integer): integer;
begin
if b = 0 then
Result := a
else Result := GCDRec(b, a mod b);
end;
begin
Println(GCD(72,30),GCDRec(72,30));
Println(GCD(49865, 69811),GCDRec(49865, 69811));
end.- Output:
6 6 9973 9973
Perl
Iterative Euclid algorithm
sub gcd_iter($$) {
my ($u, $v) = @_;
while ($v) {
($u, $v) = ($v, $u % $v);
}
return abs($u);
}Recursive Euclid algorithm
sub gcd($$) {
my ($u, $v) = @_;
if ($v) {
return gcd($v, $u % $v);
} else {
return abs($u);
}
}Iterative binary algorithm
sub gcd_bin($$) {
my ($u, $v) = @_;
$u = abs($u);
$v = abs($v);
if ($u < $v) {
($u, $v) = ($v, $u);
}
if ($v == 0) {
return $u;
}
my $k = 1;
while ($u & 1 == 0 && $v & 1 == 0) {
$u >>= 1;
$v >>= 1;
$k <<= 1;
}
my $t = ($u & 1) ? -$v : $u;
while ($t) {
while ($t & 1 == 0) {
$t >>= 1;
}
if ($t > 0) {
$u = $t;
} else {
$v = -$t;
}
$t = $u - $v;
}
return $u * $k;
}Modules
All three modules will take large integers as input, e.g. gcd("68095260063025322303723429387", "51306142182612010300800963053"). Other possibilities are Math::Cephes euclid, Math::GMPz gcd and gcd_ui.
# Fastest, takes multiple inputs
use Math::Prime::Util "gcd";
$gcd = gcd(49865, 69811);
# In CORE. Slowest, takes multiple inputs, result is a Math::BigInt unless converted
use Math::BigInt;
$gcd = Math::BigInt::bgcd(49865, 69811)->numify;
# Result is a Math::Pari object unless converted
use Math::Pari "gcd";
$gcd = gcd(49865, 69811)->pari2ivNotes on performance
use Benchmark qw(cmpthese);
use Math::BigInt;
use Math::Pari;
use Math::Prime::Util;
my $u = 40902;
my $v = 24140;
cmpthese(-5, {
'gcd_rec' => sub { gcd($u, $v); },
'gcd_iter' => sub { gcd_iter($u, $v); },
'gcd_bin' => sub { gcd_bin($u, $v); },
'gcd_bigint' => sub { Math::BigInt::bgcd($u,$v)->numify(); },
'gcd_pari' => sub { Math::Pari::gcd($u,$v)->pari2iv(); },
'gcd_mpu' => sub { Math::Prime::Util::gcd($u,$v); },
});Output on 'Intel i3930k 4.2GHz' / Linux / Perl 5.20:
Rate gcd_bigint gcd_bin gcd_rec gcd_iter gcd_pari gcd_mpu gcd_bigint 39939/s -- -83% -94% -95% -98% -99% gcd_bin 234790/s 488% -- -62% -70% -88% -97% gcd_rec 614750/s 1439% 162% -- -23% -68% -91% gcd_iter 793422/s 1887% 238% 29% -- -58% -89% gcd_pari 1896544/s 4649% 708% 209% 139% -- -73% gcd_mpu 7114798/s 17714% 2930% 1057% 797% 275% --
Phix
result is always positive, except for gcd(0,0) which is 0
atom parameters allow greater precision, but any fractional parts are immediately and deliberately discarded.
Actually, it is an autoinclude, reproduced below. The first parameter can be a sequence, in which case the second parameter (if provided) is ignored.
function gcd(object u, atom v=0) atom t if sequence(u) then v = u[1] -- (for the typecheck) t = floor(abs(v)) for i=2 to length(u) do v = u[i] -- (for the typecheck) t = gcd(t,v) end for return t end if u = floor(abs(u)) v = floor(abs(v)) while v do t = u u = v v = remainder(t, v) end while return u end function
Sample results
?gcd(0,0) -- 0 ?gcd(24,-112) -- 8 ?gcd(0, 10) -- 10 ?gcd(10, 0) -- 10 ?gcd(-10, 0) -- 10 ?gcd(0, -10) -- 10 ?gcd(9, 6) -- 3 ?gcd(6, 9) -- 3 ?gcd(-6, 9) -- 3 ?gcd(9, -6) -- 3 ?gcd(6, -9) -- 3 ?gcd(-9, 6) -- 3 ?gcd(40902, 24140) -- 34 printf(1,"%d\n",gcd(70000000000000000000, 60000000000000000000000)) -- 10000000000000000000 ?gcd({57,0,-45,-18,90,447}) -- 3
PHP
Iterative
function gcdIter($n, $m) {
while(true) {
if($n == $m) {
return $m;
}
if($n > $m) {
$n -= $m;
} else {
$m -= $n;
}
}
}Recursive
function gcdRec($n, $m)
{
if($m > 0)
return gcdRec($m, $n % $m);
else
return abs($n);
}PicoLisp
(de gcd (A B)
(until (=0 B)
(let M (% A B)
(setq A B B M) ) )
(abs A) )PL/I
GCD: procedure (a, b) returns (fixed binary (31)) recursive;
declare (a, b) fixed binary (31);
if b = 0 then return (a);
return (GCD (b, mod(a, b)) );
end GCD;Pop11
Built-in gcd
gcd_n(15, 12, 2) =>Note: the last argument gives the number of other arguments (in this case 2).
Iterative Euclid algorithm
define gcd(k, l) -> r;
lvars k , l, r = l;
abs(k) -> k;
abs(l) -> l;
if k < l then (k, l) -> (l, k) endif;
while l /= 0 do
(l, k rem l) -> (k, l)
endwhile;
k -> r;
enddefine;PostScript
/gcd {
{
{0 gt} {dup rup mod} {pop exit} ifte
} loop
}.With no external lib, recursive
/gcd {
dup 0 ne {
dup 3 1 roll mod gcd
} { pop } ifelse
} defPowerShell
Recursive Euclid Algorithm
function Get-GCD ($x, $y)
{
if ($x -eq $y) { return $y }
if ($x -gt $y) {
$a = $x
$b = $y
}
else {
$a = $y
$b = $x
}
while ($a % $b -ne 0) {
$tmp = $a % $b
$a = $b
$b = $tmp
}
return $b
}or shorter (taken from Python implementation)
function Get-GCD ($x, $y) {
if ($y -eq 0) { $x } else { Get-GCD $y ($x%$y) }
}Iterative Euclid Algorithm
based on Python implementation
Function Get-GCD( $x, $y ) {
while ($y -ne 0) {
$x, $y = $y, ($x % $y)
}
[Math]::abs($x)
}Prolog
Recursive Euclid Algorithm
gcd(X, 0, X):- !.
gcd(0, X, X):- !.
gcd(X, Y, D):- X > Y, !, Z is X mod Y, gcd(Y, Z, D).
gcd(X, Y, D):- Z is Y mod X, gcd(X, Z, D).Repeated Subtraction
gcd(X, 0, X):- !.
gcd(0, X, X):- !.
gcd(X, Y, D):- X =< Y, !, Z is Y - X, gcd(X, Z, D).
gcd(X, Y, D):- gcd(Y, X, D).Purity
data Iterate = f => FoldNat <const id, g => $g . $f>
data Sub = Iterate Pred
data IsZero = <const True, const False> . UnNat
data Eq = FoldNat
<
const IsZero,
eq => n => IfThenElse (IsZero $n)
False
($eq (Pred $n))
>
data step = gcd => n => m =>
IfThenElse (Eq $m $n)
(Pair $m $n)
(IfThenElse (Compare Leq $n $m)
($gcd (Sub $m $n) $m)
($gcd (Sub $n $m) $n))
data gcd = Iterate (gcd => uncurry (step (curry $gcd)))Python
Built-in
from fractions import gcd(Note that fractions.gcd is now deprecated in Python 3)
from math import gcdIterative Euclid algorithm
def gcd_iter(u, v):
while v:
u, v = v, u % v
return abs(u)Recursive Euclid algorithm
Interpreter: Python 2.5
def gcd(u, v):
return gcd(v, u % v) if v else abs(u)Tests
>>> gcd(0,0) 0 >>> gcd(0, 10) == gcd(10, 0) == gcd(-10, 0) == gcd(0, -10) == 10 True >>> gcd(9, 6) == gcd(6, 9) == gcd(-6, 9) == gcd(9, -6) == gcd(6, -9) == gcd(-9, 6) == 3 True >>> gcd(8, 45) == gcd(45, 8) == gcd(-45, 8) == gcd(8, -45) == gcd(-8, 45) == gcd(45, -8) == 1 True >>> gcd(40902, 24140) # check Knuth :) 34
Iterative binary algorithm
See The Art of Computer Programming by Knuth (Vol.2)
def gcd_bin(u, v):
u, v = abs(u), abs(v) # u >= 0, v >= 0
if u < v:
u, v = v, u # u >= v >= 0
if v == 0:
return u
# u >= v > 0
k = 1
while u & 1 == 0 and v & 1 == 0: # u, v - even
u >>= 1; v >>= 1
k <<= 1
t = -v if u & 1 else u
while t:
while t & 1 == 0:
t >>= 1
if t > 0:
u = t
else:
v = -t
t = u - v
return u * kNotes on performance
gcd(40902, 24140) takes us about 17 µsec (Euclid, not built-in)
gcd_iter(40902, 24140) takes us about 11 µsec
gcd_bin(40902, 24140) takes us about 41 µsec
Qi
(define gcd
A 0 -> A
A B -> (gcd B (MOD A B)))Quackery
[ [ dup while
tuck mod again ]
drop abs ] is gcd ( n n --> n )R
Recursive:
"%gcd%" <- function(u, v) {
ifelse(u %% v != 0, v %gcd% (u%%v), v)
}Iterative:
"%gcd%" <- function(v, t) {
while ( (c <- v%%t) != 0 ) {
v <- t
t <- c
}
t
}- Output:
Same either way.
> print(50 %gcd% 75) [1] 25
Racket
Racket provides a built-in gcd function. Here's a program that computes the gcd of 14 and 63:
#lang racket
(gcd 14 63)Here's an explicit implementation. Note that since Racket is tail-calling, the memory behavior of this program is "loop-like", in the sense that this program will consume no more memory than a loop-based implementation.
#lang racket
;; given two nonnegative integers, produces their greatest
;; common divisor using Euclid's algorithm
(define (gcd a b)
(if (= b 0)
a
(gcd b (modulo a b))))
;; some test cases!
(module+ test
(require rackunit)
(check-equal? (gcd (* 2 3 3 7 7)
(* 3 3 7 11))
(* 3 3 7))
(check-equal? (gcd 0 14) 14)
(check-equal? (gcd 13 0) 13))Raku
(formerly Perl 6)
Iterative
sub gcd (Int $a is copy, Int $b is copy) {
$a & $b == 0 and fail;
($a, $b) = ($b, $a % $b) while $b;
return abs $a;
}Recursive
multi gcd (0, 0) { fail }
multi gcd (Int $a, 0) { abs $a }
multi gcd (Int $a, Int $b) { gcd $b, $a % $b }Recursive(inline coderef)
{ $^b ?? &?BLOCK( $^b, $^a % $^b ) !! $^a }Concise
my &gcd = { ($^a.abs, $^b.abs, * % * ... 0)[*-2] }Actually, it's a built-in infix
my $gcd = $a gcd $b;Because it's an infix, you can use it with various meta-operators:
[gcd] @list; # reduce with gcd
@alist Zgcd @blist; # lazy zip with gcd
@alist Xgcd @blist; # lazy cross with gcd
@alist »gcd« @blist; # parallel gcdRascal
Iterative Euclidean algorithm
public int gcd_iterative(int a, b){
if(a == 0) return b;
while(b != 0){
if(a > b) a -= b;
else b -= a;}
return a;
}An example:
rascal>gcd_iterative(1989, 867)
int: 51Recursive Euclidean algorithm
public int gcd_recursive(int a, b){
return (b == 0) ? a : gcd_recursive(b, a%b);
}An example:
rascal>gcd_recursive(1989, 867)
int: 51Raven
Recursive Euclidean algorithm
define gcd use $u, $v
$v 0 > if
$u $v % $v gcd
else
$u abs
24140 40902 gcd- Output:
34
REBOL
gcd: func [
{Returns the greatest common divisor of m and n.}
m [integer!]
n [integer!]
/local k
] [
; Euclid's algorithm
while [n > 0] [
k: m
m: n
n: k // m
]
m
]Refal
$ENTRY Go {
= <Prout <Gcd 3528 3780>>;
};
Gcd {
s.M 0 = s.M;
s.M s.N = <Gcd s.N <Mod s.M s.N>>;
};- Output:
252
Retro
This is from the math extensions library.
: gcd ( ab-n ) [ tuck mod dup ] while drop ;REXX
version 1
The GCD subroutine can handle any number of arguments, it can also handle any number of integers within any
argument(s), making it easier to use when computing Frobenius numbers (also known as postage stamp or
coin numbers).
/*REXX program calculates the GCD (Greatest Common Divisor) of any number of integers.*/
numeric digits 2000 /*handle up to 2k decimal dig integers.*/
call gcd 0 0 ; call gcd 55 0 ; call gcd 0 66
call gcd 7,21 ; call gcd 41,47 ; call gcd 99 , 51
call gcd 24, -8 ; call gcd -36, 9 ; call gcd -54, -6
call gcd 14 0 7 ; call gcd 14 7 0 ; call gcd 0 14 7
call gcd 15 10 20 30 55 ; call gcd 137438691328 2305843008139952128 /*◄──2 perfect#s*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
gcd: procedure; $=; do i=1 for arg(); $=$ arg(i); end /*arg list.*/
parse var $ x z .; if x=0 then x=z; x=abs(x) /* 0 case? */
do j=2 to words($); y=abs(word($,j)); if y=0 then iterate /*is zero? */
do until _==0; _=x//y; x=y; y=_; end /* ◄────────── the heavy lifting.*/
end /*j*/
say 'GCD (Greatest Common Divisor) of ' translate(space($),",",' ') " is " x
return xoutput
GCD (Greatest Common Divisor) of 0,0 is 0 GCD (Greatest Common Divisor) of 55,0 is 55 GCD (Greatest Common Divisor) of 0,66 is 66 GCD (Greatest Common Divisor) of 7,21 is 7 GCD (Greatest Common Divisor) of 41,47 is 1 GCD (Greatest Common Divisor) of 99,51 is 3 GCD (Greatest Common Divisor) of 24,-8 is 8 GCD (Greatest Common Divisor) of -36,9 is 9 GCD (Greatest Common Divisor) of -54,-6 is 6 GCD (Greatest Common Divisor) of 14,0,7 is 7 GCD (Greatest Common Divisor) of 14,7,0 is 7 GCD (Greatest Common Divisor) of 0,14,7 is 7 GCD (Greatest Common Divisor) of 15,10,20,30,55 is 5 GCD (Greatest Common Divisor) of 137438691328,2305843008139952128 is 262144
version 2
Recursive function (as in PL/I):
/* REXX ***************************************************************
* using PL/I code extended to many arguments
* 17.08.2012 Walter Pachl
* 18.08.2012 gcd(0,0)=0
**********************************************************************/
numeric digits 300 /*handle up to 300 digit numbers.*/
Call test 7,21 ,'7 '
Call test 4,7 ,'1 '
Call test 24,-8 ,'8'
Call test 55,0 ,'55'
Call test 99,15 ,'3 '
Call test 15,10,20,30,55,'5'
Call test 496,8128 ,'16'
Call test 496,8128 ,'8' /* test wrong expectation */
Call test 0,0 ,'0' /* by definition */
Exit
test:
/**********************************************************************
* Test the gcd function
**********************************************************************/
n=arg() /* Number of arguments */
gcde=arg(n) /* Expected result */
gcdx=gcd(arg(1),arg(2)) /* gcd of the first 2 numbers */
Do i=2 To n-2 /* proceed with all the others */
If arg(i+1)<>0 Then
gcdx=gcd(gcdx,arg(i+1))
End
If gcdx=arg(arg()) Then /* result is as expected */
tag='as expected'
Else /* result is not correct */
Tag='*** wrong. expected:' gcde
numbers=arg(1) /* build string to show the input*/
Do i=2 To n-1
numbers=numbers 'and' arg(i)
End
say left('the GCD of' numbers 'is',45) right(gcdx,3) tag
Return
GCD: procedure
/**********************************************************************
* Recursive procedure as shown in PL/I
**********************************************************************/
Parse Arg a,b
if b = 0 then return abs(a)
return GCD(b,a//b)Output:
the GCD of 7 and 21 is 7 as expected the GCD of 4 and 7 is 1 as expected the GCD of 24 and -8 is 8 as expected the GCD of 55 and 0 is 55 as expected the GCD of 99 and 15 is 3 as expected the GCD of 15 and 10 and 20 and 30 and 55 is 5 as expected the GCD of 496 and 8128 is 16 as expected the GCD of 496 and 8128 is 16 *** wrong. expected: 8 the GCD of 0 and 0 is 0 as expected
version 3
using different argument handling-
Use as gcd(a,b,c,---)
Considerably faster than version 1 (and version 2)
See http://rosettacode.org/wiki/Least_common_multiple#REXX for reasoning.
gcd: procedure
x=abs(arg(1))
do j=2 to arg()
y=abs(arg(j))
If y<>0 Then Do
do until z==0
z=x//y
x=y
y=z
end
end
end
return xRing
see gcd (24, 32)
func gcd gcd, b
while b
c = gcd
gcd = b
b = c % b
end
return gcdRPL
≪ WHILE DUP REPEAT SWAP OVER MOD END DROP ABS ≫ 'GCD' STO
40902 24140 GCD
Output:
1: 34
Using unsigned integers
≪ DUP2 < ≪ SWAP ≫ IFT
WHILE DUP B→R REPEAT SWAP OVER / LAST ROT * - END DROP
≫ 'GCD' STO
#40902d #24140d GCD
Output:
1: #34d
Ruby
That is already available as the gcd method of integers:
40902.gcd(24140) # => 34Here's an implementation:
def gcd(u, v)
u, v = u.abs, v.abs
while v > 0
u, v = v, u % v
end
u
endRust
num crate
extern crate num;
use num::integer::gcd;Iterative Euclid algorithm
fn gcd(mut m: i32, mut n: i32) -> i32 {
while m != 0 {
let old_m = m;
m = n % m;
n = old_m;
}
n.abs()
}Recursive Euclid algorithm
fn gcd(m: i32, n: i32) -> i32 {
if m == 0 {
n.abs()
} else {
gcd(n % m, m)
}
}Stein's Algorithm
Stein's algorithm is very much like Euclid's except that it uses bitwise operators (and consequently slightly more performant) and the integers must be unsigned. The following is a recursive implementation that leverages Rust's pattern matching.
use std::cmp::{min, max};
fn gcd(a: usize, b: usize) -> usize {
match ((a, b), (a & 1, b & 1)) {
((x, y), _) if x == y => y,
((0, x), _) | ((x, 0), _) => x,
((x, y), (0, 1)) | ((y, x), (1, 0)) => gcd(x >> 1, y),
((x, y), (0, 0)) => gcd(x >> 1, y >> 1) << 1,
((x, y), (1, 1)) => { let (x, y) = (min(x, y), max(x, y));
gcd((y - x) >> 1, x)
}
_ => unreachable!(),
}
}Tests
println!("{}",gcd(399,-3999));
println!("{}",gcd(0,3999));
println!("{}",gcd(13*13,13*29));
3
3999
13Sass/SCSS
Iterative Euclid's Algorithm
@function gcd($a,$b) {
@while $b > 0 {
$c: $a % $b;
$a: $b;
$b: $c;
}
@return $a;
}Sather
class MATH is
gcd_iter(u, v:INT):INT is
loop while!( v.bool );
t ::= u; u := v; v := t % v;
end;
return u.abs;
end;
gcd(u, v:INT):INT is
if v.bool then return gcd(v, u%v); end;
return u.abs;
end;
private swap(inout a, inout b:INT) is
t ::= a;
a := b;
b := t;
end;
gcd_bin(u, v:INT):INT is
t:INT;
u := u.abs; v := v.abs;
if u < v then swap(inout u, inout v); end;
if v = 0 then return u; end;
k ::= 1;
loop while!( u.is_even and v.is_even );
u := u / 2; v := v / 2;
k := k * 2;
end;
if u.is_even then
t := -v;
else
t := u;
end;
loop while!( t.bool );
loop while!( t.is_even );
t := t / 2;
end;
if t > 0 then
u := t;
else
v := -t;
end;
t := u - v;
end;
return u * k;
end;
end;class MAIN is
main is
a ::= 40902;
b ::= 24140;
#OUT + MATH::gcd_iter(a, b) + "\n";
#OUT + MATH::gcd(a, b) + "\n";
#OUT + MATH::gcd_bin(a, b) + "\n";
-- built in
#OUT + a.gcd(b) + "\n";
end;
end;Scala
def gcd(a: Int, b: Int): Int = if (b == 0) a.abs else gcd(b, a % b)Using pattern matching
@tailrec
def gcd(a: Int, b: Int): Int = {
b match {
case 0 => a
case _ => gcd(b, (a % b))
}
}Scheme
(define (gcd a b)
(if (= b 0)
a
(gcd b (modulo a b))))or using the standard function included with Scheme (takes any number of arguments):
(gcd a b)Sed
#! /bin/sed -nf
# gcd.sed Copyright (c) 2010 by Paweł Zuzelski <[email protected]>
# dc.sed Copyright (c) 1995 - 1997 by Greg Ubben <[email protected]>
# usage:
#
# echo N M | ./gcd.sed
#
# Computes the greatest common divisor of N and M integers using euclidean
# algorithm.
s/^/|P|K0|I10|O10|?~/
s/$/ [lalb%sclbsalcsblb0<F]sF sasblFxlap/
:next
s/|?./|?/
s/|?#[ -}]*/|?/
/|?!*[lLsS;:<>=]\{0,1\}$/N
/|?!*[-+*/%^<>=]/b binop
/^|.*|?[dpPfQXZvxkiosStT;:]/b binop
/|?[_0-9A-F.]/b number
/|?\[/b string
/|?l/b load
/|?L/b Load
/|?[sS]/b save
/|?c/ s/[^|]*//
/|?d/ s/[^~]*~/&&/
/|?f/ s//&[pSbz0<aLb]dSaxsaLa/
/|?x/ s/\([^~]*~\)\(.*|?x\)~*/\2\1/
/|?[KIO]/ s/.*|\([KIO]\)\([^|]*\).*|?\1/\2~&/
/|?T/ s/\.*0*~/~/
# a slow, non-stackable array implementation in dc, just for completeness
# A fast, stackable, associative array implementation could be done in sed
# (format: {key}value{key}value...), but would be longer, like load & save.
/|?;/ s/|?;\([^{}]\)/|?~[s}s{L{s}q]S}[S}l\1L}1-d0>}s\1L\1l{xS\1]dS{xL}/
/|?:/ s/|?:\([^{}]\)/|?~[s}L{s}L{s}L}s\1q]S}S}S{[L}1-d0>}S}l\1s\1L\1l{xS\1]dS{x/
/|?[ ~ cdfxKIOT]/b next
/|?\n/b next
/|?[pP]/b print
/|?k/ s/^\([0-9]\{1,3\}\)\([.~].*|K\)[^|]*/\2\1/
/|?i/ s/^\(-\{0,1\}[0-9]*\.\{0,1\}[0-9]\{1,\}\)\(~.*|I\)[^|]*/\2\1/
/|?o/ s/^\(-\{0,1\}[1-9][0-9]*\.\{0,1\}[0-9]*\)\(~.*|O\)[^|]*/\2\1/
/|?[kio]/b pop
/|?t/b trunc
/|??/b input
/|?Q/b break
/|?q/b quit
h
/|?[XZz]/b count
/|?v/b sqrt
s/.*|?\([^Y]\).*/\1 is unimplemented/
s/\n/\\n/g
l
g
b next
:print
/^-\{0,1\}[0-9]*\.\{0,1\}[0-9]\{1,\}~.*|?p/!b Print
/|O10|/b Print
# Print a number in a non-decimal output base. Uses registers a,b,c,d.
# Handles fractional output bases (O<-1 or O>=1), unlike other dc's.
# Converts the fraction correctly on negative output bases, unlike
# UNIX dc. Also scales the fraction more accurately than UNIX dc.
#
s,|?p,&KSa0kd[[-]Psa0la-]Sad0>a[0P]sad0=a[A*2+]saOtd0>a1-ZSd[[[[ ]P]sclb1\
!=cSbLdlbtZ[[[-]P0lb-sb]sclb0>c1+]sclb0!<c[0P1+dld>c]scdld>cscSdLbP]q]Sb\
[t[1P1-d0<c]scd0<c]ScO_1>bO1!<cO[16]<bOX0<b[[q]sc[dSbdA>c[A]sbdA=c[B]sbd\
B=c[C]sbdC=c[D]sbdD=c[E]sbdE=c[F]sb]xscLbP]~Sd[dtdZOZ+k1O/Tdsb[.5]*[.1]O\
X^*dZkdXK-1+ktsc0kdSb-[Lbdlb*lc+tdSbO*-lb0!=aldx]dsaxLbsb]sad1!>a[[.]POX\
+sb1[SbO*dtdldx-LbO*dZlb!<a]dsax]sadXd0<asbsasaLasbLbscLcsdLdsdLdLak[]pP,
b next
:Print
/|?p/s/[^~]*/&\
~&/
s/\(.*|P\)\([^|]*\)/\
\2\1/
s/\([^~]*\)\n\([^~]*\)\(.*|P\)/\1\3\2/
h
s/~.*//
/./{ s/.//; p; }
# Just s/.//p would work if we knew we were running under the -n option.
# Using l vs p would kind of do \ continuations, but would break strings.
g
:pop
s/[^~]*~//
b next
:load
s/\(.*|?.\)\(.\)/\20~\1/
s/^\(.\)0\(.*|r\1\([^~|]*\)~\)/\1\3\2/
s/.//
b next
:Load
s/\(.*|?.\)\(.\)/\2\1/
s/^\(.\)\(.*|r\1\)\([^~|]*~\)/|\3\2/
/^|/!i\
register empty
s/.//
b next
:save
s/\(.*|?.\)\(.\)/\2\1/
/^\(.\).*|r\1/ !s/\(.\).*|/&r\1|/
/|?S/ s/\(.\).*|r\1/&~/
s/\(.\)\([^~]*~\)\(.*|r\1\)[^~|]*~\{0,1\}/\3\2/
b next
:quit
t quit
s/|?[^~]*~[^~]*~/|?q/
t next
# Really should be using the -n option to avoid printing a final newline.
s/.*|P\([^|]*\).*/\1/
q
:break
s/[0-9]*/&;987654321009;/
:break1
s/^\([^;]*\)\([1-9]\)\(0*\)\([^1]*\2\(.\)[^;]*\3\(9*\).*|?.\)[^~]*~/\1\5\6\4/
t break1
b pop
:input
N
s/|??\(.*\)\(\n.*\)/|?\2~\1/
b next
:count
/|?Z/ s/~.*//
/^-\{0,1\}[0-9]*\.\{0,1\}[0-9]\{1,\}$/ s/[-.0]*\([^.]*\)\.*/\1/
/|?X/ s/-*[0-9A-F]*\.*\([0-9A-F]*\).*/\1/
s/|.*//
/~/ s/[^~]//g
s/./a/g
:count1
s/a\{10\}/b/g
s/b*a*/&a9876543210;/
s/a.\{9\}\(.\).*;/\1/
y/b/a/
/a/b count1
G
/|?z/ s/\n/&~/
s/\n[^~]*//
b next
:trunc
# for efficiency, doesn't pad with 0s, so 10k 2 5/ returns just .40
# The X* here and in a couple other places works around a SunOS 4.x sed bug.
s/\([^.~]*\.*\)\(.*|K\([^|]*\)\)/\3;9876543210009909:\1,\2/
:trunc1
s/^\([^;]*\)\([1-9]\)\(0*\)\([^1]*\2\(.\)[^:]*X*\3\(9*\)[^,]*\),\([0-9]\)/\1\5\6\4\7,/
t trunc1
s/[^:]*:\([^,]*\)[^~]*/\1/
b normal
:number
s/\(.*|?\)\(_\{0,1\}[0-9A-F]*\.\{0,1\}[0-9A-F]*\)/\2~\1~/
s/^_/-/
/^[^A-F~]*~.*|I10|/b normal
/^[-0.]*~/b normal
s:\([^.~]*\)\.*\([^~]*\):[Ilb^lbk/,\1\2~0A1B2C3D4E5F1=11223344556677889900;.\2:
:digit
s/^\([^,]*\),\(-*\)\([0-F]\)\([^;]*\(.\)\3[^1;]*\(1*\)\)/I*+\1\2\6\5~,\2\4/
t digit
s:...\([^/]*.\)\([^,]*\)[^.]*\(.*|?.\):\2\3KSb[99]k\1]SaSaXSbLalb0<aLakLbktLbk:
b next
:string
/|?[^]]*$/N
s/\(|?[^]]*\)\[\([^]]*\)]/\1|{\2|}/
/|?\[/b string
s/\(.*|?\)|{\(.*\)|}/\2~\1[/
s/|{/[/g
s/|}/]/g
b next
:binop
/^[^~|]*~[^|]/ !i\
stack empty
//!b next
/^-\{0,1\}[0-9]*\.\{0,1\}[0-9]\{1,\}~/ !s/[^~]*\(.*|?!*[^!=<>]\)/0\1/
/^[^~]*~-\{0,1\}[0-9]*\.\{0,1\}[0-9]\{1,\}~/ !s/~[^~]*\(.*|?!*[^!=<>]\)/~0\1/
h
/|?\*/b mul
/|?\//b div
/|?%/b rem
/|?^/b exp
/|?[+-]/ s/^\(-*\)\([^~]*~\)\(-*\)\([^~]*~\).*|?\(-\{0,1\}\).*/\2\4s\3o\1\3\5/
s/\([^.~]*\)\([^~]*~[^.~]*\)\(.*\)/<\1,\2,\3|=-~.0,123456789<></
/^<\([^,]*,[^~]*\)\.*0*~\1\.*0*~/ s/</=/
:cmp1
s/^\(<[^,]*\)\([0-9]\),\([^,]*\)\([0-9]\),/\1,\2\3,\4/
t cmp1
/^<\([^~]*\)\([^~]\)[^~]*~\1\(.\).*|=.*\3.*\2/ s/</>/
/|?/{
s/^\([<>]\)\(-[^~]*~-.*\1\)\(.\)/\3\2/
s/^\(.\)\(.*|?!*\)\1/\2!\1/
s/|?![^!]\(.\)/&l\1x/
s/[^~]*~[^~]*~\(.*|?\)!*.\(.*\)|=.*/\1\2/
b next
}
s/\(-*\)\1|=.*/;9876543210;9876543210/
/o-/ s/;9876543210/;0123456789/
s/^>\([^~]*~\)\([^~]*~\)s\(-*\)\(-*o\3\(-*\)\)/>\2\1s\5\4/
s/,\([0-9]*\)\.*\([^,]*\),\([0-9]*\)\.*\([0-9]*\)/\1,\2\3.,\4;0/
:right1
s/,\([0-9]\)\([^,]*\),;*\([0-9]\)\([0-9]*\);*0*/\1,\2\3,\4;0/
t right1
s/.\([^,]*\),~\(.*\);0~s\(-*\)o-*/\1~\30\2~/
:addsub1
s/\(.\{0,1\}\)\(~[^,]*\)\([0-9]\)\(\.*\),\([^;]*\)\(;\([^;]*\(\3[^;]*\)\).*X*\1\(.*\)\)/\2,\4\5\9\8\7\6/
s/,\([^~]*~\).\{10\}\(.\)[^;]\{0,9\}\([^;]\{0,1\}\)[^;]*/,\2\1\3/
# could be done in one s/// if we could have >9 back-refs...
/^~.*~;/!b addsub1
:endbin
s/.\([^,]*\),\([0-9.]*\).*/\1\2/
G
s/\n[^~]*~[^~]*//
:normal
s/^\(-*\)0*\([0-9.]*[0-9]\)[^~]*/\1\2/
s/^[^1-9~]*~/0~/
b next
:mul
s/\(-*\)\([0-9]*\)\.*\([0-9]*\)~\(-*\)\([0-9]*\)\.*\([0-9]*\).*|K\([^|]*\).*/\1\4\2\5.!\3\6,|\2<\3~\5>\6:\7;9876543210009909/
:mul1
s/![0-9]\([^<]*\)<\([0-9]\{0,1\}\)\([^>]*\)>\([0-9]\{0,1\}\)/0!\1\2<\3\4>/
/![0-9]/ s/\(:[^;]*\)\([1-9]\)\(0*\)\([^0]*\2\(.\).*X*\3\(9*\)\)/\1\5\6\4/
/<~[^>]*>:0*;/!t mul1
s/\(-*\)\1\([^>]*\).*/;\2^>:9876543210aaaaaaaaa/
:mul2
s/\([0-9]~*\)^/^\1/
s/<\([0-9]*\)\(.*[~^]\)\([0-9]*\)>/\1<\2>\3/
:mul3
s/>\([0-9]\)\(.*\1.\{9\}\(a*\)\)/\1>\2;9\38\37\36\35\34\33\32\31\30/
s/\(;[^<]*\)\([0-9]\)<\([^;]*\).*\2[0-9]*\(.*\)/\4\1<\2\3/
s/a[0-9]/a/g
s/a\{10\}/b/g
s/b\{10\}/c/g
/|0*[1-9][^>]*>0*[1-9]/b mul3
s/;/a9876543210;/
s/a.\{9\}\(.\)[^;]*\([^,]*\)[0-9]\([.!]*\),/\2,\1\3/
y/cb/ba/
/|<^/!b mul2
b endbin
:div
# CDDET
/^[-.0]*[1-9]/ !i\
divide by 0
//!b pop
s/\(-*\)\([0-9]*\)\.*\([^~]*~-*\)\([0-9]*\)\.*\([^~]*\)/\2.\3\1;0\4.\5;0/
:div1
s/^\.0\([^.]*\)\.;*\([0-9]\)\([0-9]*\);*0*/.\1\2.\3;0/
s/^\([^.]*\)\([0-9]\)\.\([^;]*;\)0*\([0-9]*\)\([0-9]\)\./\1.\2\30\4.\5/
t div1
s/~\(-*\)\1\(-*\);0*\([^;]*[0-9]\)[^~]*/~123456789743222111~\2\3/
s/\(.\(.\)[^~]*\)[^9]*\2.\{8\}\(.\)[^~]*/\3~\1/
s,|?.,&SaSadSaKdlaZ+LaX-1+[sb1]Sbd1>bkLatsbLa[dSa2lbla*-*dLa!=a]dSaxsakLasbLb*t,
b next
:rem
s,|?%,&Sadla/LaKSa[999]k*Lak-,
b next
:exp
# This decimal method is just a little faster than the binary method done
# totally in dc: 1LaKLb [kdSb*LbK]Sb [[.5]*d0ktdSa<bkd*KLad1<a]Sa d1<a kk*
/^[^~]*\./i\
fraction in exponent ignored
s,[^-0-9].*,;9d**dd*8*d*d7dd**d*6d**d5d*d*4*d3d*2lbd**1lb*0,
:exp1
s/\([0-9]\);\(.*\1\([d*]*\)[^l]*\([^*]*\)\(\**\)\)/;dd*d**d*\4\3\5\2/
t exp1
G
s,-*.\{9\}\([^9]*\)[^0]*0.\(.*|?.\),\2~saSaKdsaLb0kLbkK*+k1\1LaktsbkLax,
s,|?.,&SadSbdXSaZla-SbKLaLadSb[0Lb-d1lb-*d+K+0kkSb[1Lb/]q]Sa0>a[dk]sadK<a[Lb],
b next
:sqrt
# first square root using sed: 8k2v at 1:30am Dec 17, 1996
/^-/i\
square root of negative number
/^[-0]/b next
s/~.*//
/^\./ s/0\([0-9]\)/\1/g
/^\./ !s/[0-9][0-9]/7/g
G
s/\n/~/
s,|?.,&K1+k KSbSb[dk]SadXdK<asadlb/lb+[.5]*[sbdlb/lb+[.5]*dlb>a]dsaxsasaLbsaLatLbk K1-kt,
b next
# END OF GSU dc.sedSeed7
const func integer: gcd (in var integer: a, in var integer: b) is func
result
var integer: gcd is 0;
local
var integer: help is 0;
begin
while a <> 0 do
help := b rem a;
b := a;
a := help;
end while;
gcd := b;
end func;Original source: [1]
SequenceL
Tail Recursive Greatest Common Denominator using Euclidian Algorithm
gcd(a, b) :=
a when b = 0
else
gcd(b, a mod b);SETL
a := 33; b := 77;
print(" the gcd of",a," and ",b," is ",gcd(a,b));
c := 49865; d := 69811;
print(" the gcd of",c," and ",d," is ",gcd(c,d));
proc gcd (u, v);
return if v = 0 then abs u else gcd (v, u mod v) end;
end;Output:
the gcd of 33 and 77 is 11
the gcd of 49865 and 69811 is 9973Sidef
Built-in
var arr = [100, 1_000, 10_000, 20];
say Math.gcd(arr...);Recursive Euclid algorithm
func gcd(a, b) {
b.is_zero ? a.abs : gcd(b, a % b);
}Simula
For a recursive variant, see Sum multiples of 3 and 5.
BEGIN
INTEGER PROCEDURE GCD(a, b); INTEGER a, b;
BEGIN
IF a = 0 THEN a := b
ELSE
WHILE 0 < b DO BEGIN INTEGER i;
i := MOD(a, b); a := b; b := i;
END;
GCD := a
END;
INTEGER a, b;
!outint(SYSOUT.IMAGE.MAIN.LENGTH, 0);!OUTIMAGE;!OUTIMAGE;
!SYSOUT.IMAGE :- BLANKS(132); ! this may or may not work;
FOR b := 1 STEP 5 UNTIL 37 DO BEGIN
FOR a := 0 STEP 2 UNTIL 21 DO BEGIN
OUTTEXT(" ("); OUTINT(a, 0);
OUTCHAR(','); OUTINT(b, 2);
OUTCHAR(')'); OUTINT(GCD(a, b), 3);
END;
OUTIMAGE
END
END- Output:
(0, 1) 1 (2, 1) 1 (4, 1) 1 (6, 1) 1 (8, 1) 1 (10, 1) 1 (12, 1) 1 (14, 1) 1 (16, 1) 1 (18, 1) 1 (20, 1) 1 (0, 6) 6 (2, 6) 2 (4, 6) 2 (6, 6) 6 (8, 6) 2 (10, 6) 2 (12, 6) 6 (14, 6) 2 (16, 6) 2 (18, 6) 6 (20, 6) 2 (0,11) 11 (2,11) 1 (4,11) 1 (6,11) 1 (8,11) 1 (10,11) 1 (12,11) 1 (14,11) 1 (16,11) 1 (18,11) 1 (20,11) 1 (0,16) 16 (2,16) 2 (4,16) 4 (6,16) 2 (8,16) 8 (10,16) 2 (12,16) 4 (14,16) 2 (16,16) 16 (18,16) 2 (20,16) 4 (0,21) 21 (2,21) 1 (4,21) 1 (6,21) 3 (8,21) 1 (10,21) 1 (12,21) 3 (14,21) 7 (16,21) 1 (18,21) 3 (20,21) 1 (0,26) 26 (2,26) 2 (4,26) 2 (6,26) 2 (8,26) 2 (10,26) 2 (12,26) 2 (14,26) 2 (16,26) 2 (18,26) 2 (20,26) 2 (0,31) 31 (2,31) 1 (4,31) 1 (6,31) 1 (8,31) 1 (10,31) 1 (12,31) 1 (14,31) 1 (16,31) 1 (18,31) 1 (20,31) 1 (0,36) 36 (2,36) 2 (4,36) 4 (6,36) 6 (8,36) 4 (10,36) 2 (12,36) 12 (14,36) 2 (16,36) 4 (18,36) 18 (20,36) 4
Slate
Slate's Integer type has gcd defined:
40902 gcd: 24140Iterative Euclid algorithm
x@(Integer traits) gcd: y@(Integer traits)
"Euclid's algorithm for finding the greatest common divisor."
[| n m temp |
n: x.
m: y.
[n isZero] whileFalse: [temp: n. n: m \\ temp. m: temp].
m abs
].Recursive Euclid algorithm
x@(Integer traits) gcd: y@(Integer traits)
[
y isZero
ifTrue: [x]
ifFalse: [y gcd: x \\ y]
].Smalltalk
The Integer class has its gcd method.
(40902 gcd: 24140) displayNlAn reimplementation of the Iterative Euclid's algorithm would be:
|gcd_iter|
gcd_iter := [ :a :b |
|u v|
u := a. v := b.
[ v > 0 ]
whileTrue: [ |t|
t := u.
u := v.
v := t rem: v
].
u abs
].
(gcd_iter value: 40902 value: 24140) printNl.SNOBOL4
define('gcd(i,j)') :(gcd_end)
gcd ?eq(i,0) :s(freturn)
?eq(j,0) :s(freturn)
loop gcd = remdr(i,j)
gcd = ?eq(gcd,0) j :s(return)
i = j
j = gcd :(loop)
gcd_end
output = gcd(1071,1029)
endSparkling
function factors(n) {
var f = {};
for var i = 2; n > 1; i++ {
while n % i == 0 {
n /= i;
f[i] = f[i] != nil ? f[i] + 1 : 1;
}
}
return f;
}
function GCD(n, k) {
let f1 = factors(n);
let f2 = factors(k);
let fs = map(f1, function(factor, multiplicity) {
let m = f2[factor];
return m == nil ? 0 : min(m, multiplicity);
});
let rfs = {};
foreach(fs, function(k, v) {
rfs[sizeof rfs] = pow(k, v);
});
return reduce(rfs, 1, function(x, y) { return x * y; });
}
function LCM(n, k) {
return n * k / GCD(n, k);
}SQL
Demonstration of Oracle 12c WITH Clause Enhancements
drop table tbl;
create table tbl
(
u number,
v number
);
insert into tbl ( u, v ) values ( 20, 50 );
insert into tbl ( u, v ) values ( 21, 50 );
insert into tbl ( u, v ) values ( 21, 51 );
insert into tbl ( u, v ) values ( 22, 50 );
insert into tbl ( u, v ) values ( 22, 55 );
commit;
with
function gcd ( ui in number, vi in number )
return number
is
u number := ui;
v number := vi;
t number;
begin
while v > 0
loop
t := u;
u := v;
v:= mod(t, v );
end loop;
return abs(u);
end gcd;
select u, v, gcd ( u, v )
from tbl
/- Output:
Table dropped.
Table created.
1 row created.
1 row created.
1 row created.
1 row created.
1 row created.
Commit complete.
U V GCD(U,V)
---------- ---------- ----------
20 50 10
21 50 1
21 51 3
22 50 2
22 55 11
Demonstration of SQL Server 2008
CREATE FUNCTION gcd (
@ui INT,
@vi INT
) RETURNS INT
AS
BEGIN
DECLARE @t INT
DECLARE @u INT
DECLARE @v INT
SET @u = @ui
SET @v = @vi
WHILE @v > 0
BEGIN
SET @t = @u;
SET @u = @v;
SET @v = @t % @v;
END;
RETURN abs( @u );
END
GO
CREATE TABLE tbl (
u INT,
v INT
);
INSERT INTO tbl ( u, v ) VALUES ( 20, 50 );
INSERT INTO tbl ( u, v ) VALUES ( 21, 50 );
INSERT INTO tbl ( u, v ) VALUES ( 21, 51 );
INSERT INTO tbl ( u, v ) VALUES ( 22, 50 );
INSERT INTO tbl ( u, v ) VALUES ( 22, 55 );
SELECT u, v, dbo.gcd ( u, v )
FROM tbl;
DROP TABLE tbl;
DROP FUNCTION gcd;PostgreSQL function using a recursive common table expression
CREATE FUNCTION gcd(integer, integer)
RETURNS integer
LANGUAGE sql
AS $function$
WITH RECURSIVE x (u, v) AS (
SELECT ABS($1), ABS($2)
UNION
SELECT v, u % v FROM x WHERE v > 0
)
SELECT min(u) FROM x;
$function$- Output:
postgres> select gcd(40902, 24140); gcd ----- 34 SELECT 1 Time: 0.012s
Standard ML
See also #ML / Standard ML.
(* Euclid’s algorithm. *)
fun gcd (u, v) =
let
fun loop (u, v) =
if v = 0 then
u
else
loop (v, u mod v)
in
loop (abs u, abs v)
end
(* Using the Rosetta Code example for assertions in Standard ML. *)
fun assert cond =
if cond then () else raise Fail "assert"
val () = assert (gcd (0, 0) = 0)
val () = assert (gcd (0, 10) = 10)
val () = assert (gcd (~10, 0) = 10)
val () = assert (gcd (9, 6) = 3)
val () = assert (gcd (~6, ~9) = 3)
val () = assert (gcd (40902, 24140) = 34)
val () = assert (gcd (40902, ~24140) = 34)
val () = assert (gcd (~40902, 24140) = 34)
val () = assert (gcd (~40902, ~24140) = 34)
val () = assert (gcd (24140, 40902) = 34)
val () = assert (gcd (~24140, 40902) = 34)
val () = assert (gcd (24140, ~40902) = 34)
val () = assert (gcd (~24140, ~40902) = 34)Stata
function gcd(a_,b_) {
a = abs(a_)
b = abs(b_)
while (b>0) {
a = mod(a,b)
swap(a,b)
}
return(a)
}Swift
// Iterative
func gcd(a: Int, b: Int) -> Int {
var a = abs(a)
var b = abs(b)
if (b > a) { swap(&a, &b) }
while (b > 0) { (a, b) = (b, a % b) }
return a
}
// Recursive
func gcdr (a: Int, b: Int) -> Int {
var a = abs(a)
var b = abs(b)
if (b > a) { swap(&a, &b) }
return gcd_rec(a,b)
}
private func gcd_rec(a: Int, b: Int) -> Int {
return b == 0 ? a : gcd_rec(b, a % b)
}
for (a,b) in [(1,1), (100, -10), (10, -100), (-36, -17), (27, 18), (30, -42)] {
println("Iterative: GCD of \(a) and \(b) is \(gcd(a, b))")
println("Recursive: GCD of \(a) and \(b) is \(gcdr(a, b))")
}- Output:
Iterative: GCD of 1 and 1 is 1 Recursive: GCD of 1 and 1 is 1 Iterative: GCD of 100 and -10 is 10 Recursive: GCD of 100 and -10 is 10 Iterative: GCD of 10 and -100 is 10 Recursive: GCD of 10 and -100 is 10 Iterative: GCD of -36 and -17 is 1 Recursive: GCD of -36 and -17 is 1 Iterative: GCD of 27 and 18 is 9 Recursive: GCD of 27 and 18 is 9 Iterative: GCD of 30 and -42 is 6 Recursive: GCD of 30 and -42 is 6
Tcl
Iterative Euclid algorithm
package require Tcl 8.5
namespace path {::tcl::mathop ::tcl::mathfunc}
proc gcd_iter {p q} {
while {$q != 0} {
lassign [list $q [% $p $q]] p q
}
abs $p
}Recursive Euclid algorithm
proc gcd {p q} {
if {$q == 0} {
return $p
}
gcd $q [expr {$p % $q}]
}With Tcl 8.6, this can be optimized slightly to:
proc gcd {p q} {
if {$q == 0} {
return $p
}
tailcall gcd $q [expr {$p % $q}]
}(Tcl does not perform automatic tail-call optimization introduction because that makes any potential error traces less informative.)
Iterative binary algorithm
package require Tcl 8.5
namespace path {::tcl::mathop ::tcl::mathfunc}
proc gcd_bin {p q} {
if {$p == $q} {return [abs $p]}
set p [abs $p]
if {$q == 0} {return $p}
set q [abs $q]
if {$p < $q} {lassign [list $q $p] p q}
set k 1
while {($p & 1) == 0 && ($q & 1) == 0} {
set p [>> $p 1]
set q [>> $q 1]
set k [<< $k 1]
}
set t [expr {$p & 1 ? -$q : $p}]
while {$t} {
while {$t & 1 == 0} {set t [>> $t 1]}
if {$t > 0} {set p $t} {set q [- $t]}
set t [- $p $q]
}
return [* $p $k]
}Notes on performance
foreach proc {gcd_iter gcd gcd_bin} {
puts [format "%-8s - %s" $proc [time {$proc $u $v} 100000]]
}Outputs:
gcd_iter - 4.46712 microseconds per iteration gcd - 5.73969 microseconds per iteration gcd_bin - 9.25613 microseconds per iteration
Transact-SQL
CREATE OR ALTER FUNCTION [dbo].[PGCD]
( @a BigInt
, @b BigInt
)
RETURNS BigInt
WITH RETURNS NULL ON NULL INPUT
-- Calculates the Greatest Common Denominator of two numbers (1 if they are coprime).
BEGIN
DECLARE @PGCD BigInt;
WITH Vars(A, B)
As ( SELECT Max(V.N) As A
, Min(V.N) As B
FROM ( VALUES ( Abs(@a) , Abs(@b)) ) Params(A, B)
-- First, get absolute value
Cross APPLY ( VALUES (Params.A) , (Params.B) ) V(N)
-- Then, order parameters without Greatest/Least functions
WHERE Params.A > 0
And Params.B > 0 -- If 0 passed in, NULL shall be the output
)
, Calc(A, B)
As ( SELECT A
, B
FROM Vars
UNION ALL
SELECT B As A
, A % B As B -- Self-ordering
FROM Calc
WHERE Calc.A > 0
And Calc.B > 0
)
SELECT @PGCD = Min(A)
FROM Calc
WHERE Calc.B = 0
;
RETURN @PGCD;
ENDTSE SAL
// library: math: get: greatest: common: divisor <description>greatest common divisor whole numbers. Euclid's algorithm. Recursive version</description> <version control></version control> <version>1.0.0.0.3</version> <version control></version control> (filenamemacro=getmacdi.s) [<Program>] [<Research>] [kn, ri, su, 20-01-2013 14:22:41]
INTEGER PROC FNMathGetGreatestCommonDivisorI( INTEGER x1I, INTEGER x2I )
//
IF ( x2I == 0 )
//
RETURN( x1I )
//
ENDIF
//
RETURN( FNMathGetGreatestCommonDivisorI( x2I, x1I MOD x2I ) )
//
END
PROC Main()
STRING s1[255] = "353"
STRING s2[255] = "46"
REPEAT
IF ( NOT ( Ask( " = ", s1, _EDIT_HISTORY_ ) ) AND ( Length( s1 ) > 0 ) ) RETURN() ENDIF
IF ( NOT ( Ask( " = ", s2, _EDIT_HISTORY_ ) ) AND ( Length( s2 ) > 0 ) ) RETURN() ENDIF
Warn( FNMathGetGreatestCommonDivisorI( Val( s1 ), Val( s2 ) ) ) // gives e.g. 1
UNTIL FALSE
ENDTXR
$ txr -p '(gcd (expt 2 123) (expt 6 49))'
562949953421312TypeScript
Iterative implementation
function gcd(a: number, b: number) {
a = Math.abs(a);
b = Math.abs(b);
if (b > a) {
let temp = a;
a = b;
b = temp;
}
while (true) {
a %= b;
if (a === 0) { return b; }
b %= a;
if (b === 0) { return a; }
}
}Recursive.
function gcd_rec(a: number, b: number) {
return b ? gcd_rec(b, a % b) : Math.abs(a);
}Uiua
⊙◌⍢(⊃∘◿:)(±,)UNIX Shell
gcd() {
# Calculate $1 % $2 until $2 becomes zero.
until test 0 -eq "$2"; do
# Parallel assignment: set -- 1 2
set -- "$2" "`expr "$1" % "$2"`"
done
# Echo absolute value of $1.
test 0 -gt "$1" && set -- "`expr 0 - "$1"`"
echo "$1"
}
gcd -47376 87843
# => 987dash or bash
Procedural :
gcd() { until test 0 -eq "$2";do set -- "$2" "$(($1 % $2))";done;if [ 0 -gt "$1" ];then echo "$((- $1))";else echo "$1"; fi }
gcd -47376 87843
# => 987
Recursive :
gcd () { if [ "$2" -ne 0 ];then gcd "$2" "$(($1 % $2))";else echo "$1";fi }
gcd 100 75
# => 25C Shell
alias gcd eval \''set gcd_args=( \!*:q ) \\
@ gcd_u=$gcd_args[2] \\
@ gcd_v=$gcd_args[3] \\
while ( $gcd_v != 0 ) \\
@ gcd_t = $gcd_u % $gcd_v \\
@ gcd_u = $gcd_v \\
@ gcd_v = $gcd_t \\
end \\
if ( $gcd_u < 0 ) @ gcd_u = - $gcd_u \\
@ $gcd_args[1]=$gcd_u \\
'\'
gcd result -47376 87843
echo $result
# => 987Ursa
import "math"
out (gcd 40902 24140) endl console- Output:
34
Ursala
This doesn't need to be defined because it's a library function, but it can be defined like this based on a recursive implementation of Euclid's algorithm. This isn't the simplest possible solution because it includes a bit shifting optimization that happens when both operands are even.
#import nat
gcd = ~&B?\~&Y ~&alh^?\~&arh2faltPrXPRNfabt2RCQ @a ~&ar^?\~&al ^|R/~& ^/~&r remaindertest program:
#cast %nWnAL
test = ^(~&,gcd)* <(25,15),(36,16),(120,45),(30,100)>output:
< (25,15): 5, (36,16): 4, (120,45): 15, (30,100): 10>
V
like joy
iterative
[gcd [0 >] [dup rollup %] while pop ].
recursive
like python
[gcd
[zero?] [pop]
[swap [dup] dip swap %]
tailrec].
same with view: (swap [dup] dip swap % is replaced with a destructuring view)
[gcd
[zero?] [pop]
[[a b : [b a b %]] view i]
tailrec].
running it
|1071 1029 gcd =21
Verilog
module gcd
(
input reset_l,
input clk,
input [31:0] initial_u,
input [31:0] initial_v,
input load,
output reg [31:0] result,
output reg busy
);
reg [31:0] u, v;
always @(posedge clk or negedge reset_l)
if (!reset_l)
begin
busy <= 0;
u <= 0;
v <= 0;
end
else
begin
result <= u + v; // Result (one of them will be zero)
busy <= u && v; // We're still busy...
// Repeatedly subtract smaller number from larger one
if (v <= u)
u <= u - v;
else if (u < v)
v <= v - u;
if (load) // Load new problem when high
begin
u <= initial_u;
v <= initial_v;
busy <= 1;
end
end
endmoduleV (Vlang)
Iterative
fn gcd(xx int, yy int) int {
mut x, mut y := xx, yy
for y != 0 {
x, y = y, x%y
}
return x
}
fn main() {
println(gcd(33, 77))
println(gcd(49865, 69811))
}Builtin
(This is just a wrapper for big.gcd)
import math.big
fn gcd(x i64, y i64) i64 {
return big.integer_from_i64(x).gcd(big.integer_from_i64(y)).int()
}
fn main() {
println(gcd(33, 77))
println(gcd(49865, 69811))
}- Output in either case:
11 9973
Wortel
Operator
@gcd a bNumber expression
!#~kg a bIterative
&[a b] [@vars[t] @while b @:{t b b %a b a t} a]Recursive
&{gcd a b} ?{b !!gcd b %a b @abs a}Wren
var gcd = Fn.new { |x, y|
while (y != 0) {
var t = y
y = x % y
x = t
}
return x.abs
}
System.print("gcd(33, 77) = %(gcd.call(33, 77))")
System.print("gcd(49865, 69811) = %(gcd.call(49865, 69811))")- Output:
gcd(33, 77) = 11 gcd(49865, 69811) = 9973
x86 Assembly
Using GNU Assembler syntax:
.text
.global pgcd
pgcd:
push %ebp
mov %esp, %ebp
mov 8(%ebp), %eax
mov 12(%ebp), %ecx
push %edx
.loop:
cmp $0, %ecx
je .end
xor %edx, %edx
div %ecx
mov %ecx, %eax
mov %edx, %ecx
jmp .loop
.end:
pop %edx
leave
retXLISP
GCD is a built-in function. If we wanted to reimplement it, one (tail-recursive) way would be like this:
(defun greatest-common-divisor (x y)
(if (= y 0)
x
(greatest-common-divisor y (mod x y)) ) )XPL0
include c:\cxpl\codes;
func GCD(U, V); \Return the greatest common divisor of U and V
int U, V;
int T;
[while V do \Euclid's method
[T:= U; U:= V; V:= rem(T/V)];
return abs(U);
];
\Display the GCD of two integers entered on command line
IntOut(0, GCD(IntIn(8), IntIn(8)))YAMLScript
!yamlscript/v0
defn main(a=42 b=63):
say: "gcd($a $b) -> $gcd(a b)"
defn gcd(a b):
if b > 0:
recur b: a % b
else: aZ80 Assembly
Uses the iterative subtraction implementation of Euclid's algorithm because the Z80 does not implement modulus or division opcodes.
; Inputs: a, b
; Outputs: a = gcd(a, b)
; Destroys: c
; Assumes: a and b are positive one-byte integers
gcd:
cp b
ret z ; while a != b
jr c, else ; if a > b
sub b ; a = a - b
jr gcd
else:
ld c, a ; Save a
ld a, b ; Swap b into a so we can do the subtraction
sub c ; b = b - a
ld b, a ; Put a and b back where they belong
ld a, c
jr gcdZig
pub fn gcd(u: anytype, v: anytype) @TypeOf(u) {
if (@typeInfo(@TypeOf(u)) != .Int) {
@compileError("non-integer type used on gcd: " ++ @typeName(@TypeOf(u)));
}
if (@typeInfo(@TypeOf(v)) != .Int) {
@compileError("non-integer type used on gcd: " ++ @typeName(@TypeOf(v)));
}
return if (v != 0) gcd(v, @mod(u,v)) else u;
}zkl
This is a method on integers:
(123456789).gcd(987654321) //-->9Using the gnu big num library (GMP):
var BN=Import("zklBigNum");
BN(123456789).gcd(987654321) //-->9or
fcn gcd(a,b){ while(b){ t:=a; a=b; b=t%b } a.abs() }Categories:
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