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For positive integer arguments, the GCD is the largest positive number which divides both numbers. If one of the arguments is zero, the GCD function returns the other number. | For positive integer arguments, the GCD is the largest positive number which divides both numbers. If one of the arguments is zero, the GCD function returns the other number. | ||
< | <syntaxhighlight lang=apl> | ||
∘.∨⍨ 0,⍳10 | ∘.∨⍨ 0,⍳10 | ||
0 1 2 3 4 5 6 7 8 9 10 | 0 1 2 3 4 5 6 7 8 9 10 | ||
| Line 18: | Line 18: | ||
9 1 1 3 1 1 3 1 1 9 1 | 9 1 1 3 1 1 3 1 1 9 1 | ||
10 1 2 1 2 5 2 1 2 1 10 | 10 1 2 1 2 5 2 1 2 1 10 | ||
</ | </syntaxhighlight>{{Works in|[[Dyalog APL]]}} | ||
While the mathematical definition of GCD does not cover non-integers, some implementations accept them as arguments. In this case, the return value of < | While the mathematical definition of GCD does not cover non-integers, some implementations accept them as arguments. In this case, the return value of <syntaxhighlight lang=apl inline>R←X∨Y</syntaxhighlight> is chosen so that both <syntaxhighlight lang=apl inline>X÷R</syntaxhighlight> and <syntaxhighlight lang=apl inline>Y÷R</syntaxhighlight> are integers (or [[wikipedia:Gaussian integer|Gaussian integers]], when X and/or Y are [[complex]] numbers). | ||
< | <syntaxhighlight lang=apl> | ||
0.6∨13÷3 | 0.6∨13÷3 | ||
0.06666666667 | 0.06666666667 | ||
| Line 31: | Line 31: | ||
2J2 3J1÷1J1 | 2J2 3J1÷1J1 | ||
2 2J¯1 | 2 2J¯1 | ||
</ | </syntaxhighlight>{{Works in|[[Dyalog APL]]}} | ||
== Description == | == Description == | ||
If both arguments are integers, the GCD is the greatest positive integer which divides both numbers evenly, or 0 if both arguments are 0. That is, each argument is an integer multiple of the GCD of both arguments. Under this definition, any number is considered a divisor of 0, because multiplying it by 0 results in 0. Using [[Residue]], we might also write the divisibility criterion as < | If both arguments are integers, the GCD is the greatest positive integer which divides both numbers evenly, or 0 if both arguments are 0. That is, each argument is an integer multiple of the GCD of both arguments. Under this definition, any number is considered a divisor of 0, because multiplying it by 0 results in 0. Using [[Residue]], we might also write the divisibility criterion as <syntaxhighlight lang=apl inline>0 0≡(a∨b)|a,b</syntaxhighlight>. | ||
Because 1 divides every integer, there is always some common divisor of any pair of arguments, and the GCD is well-defined. The [[identity element]] for GCD is 0 on the domain of non-negative real numbers, because the other argument will always be a divisor of 0, and so it is returned as the result (for an arbitrary real number, the result is its [[absolute value]]). Because 1 is the only positive divisor of itself, the GCD of 1 and any other number is 1. | Because 1 divides every integer, there is always some common divisor of any pair of arguments, and the GCD is well-defined. The [[identity element]] for GCD is 0 on the domain of non-negative real numbers, because the other argument will always be a divisor of 0, and so it is returned as the result (for an arbitrary real number, the result is its [[absolute value]]). Because 1 is the only positive divisor of itself, the GCD of 1 and any other number is 1. | ||
| Line 41: | Line 41: | ||
== History == | == History == | ||
The use of GCD as an extension of [[Or]], and its extension to complex rational numbers, was proposed by [[Eugene McDonnell]] at [[APL75]].<ref>[[Eugene McDonnell]]. [https://doi.org/10.1145/800117.803810 "A Notation for the GCD and LCM Functions"] ([https://www.jsoftware.com/papers/eem/gcd.htm web]) at [[APL75]].</ref> This definition has become common among many APLs, with [[SHARP APL]], [[Dyalog APL]] (as of [[Dyalog APL 11.0|version 11.0]]), [[J]], [[NARS2000]], [[GNU APL]], [[ngn/apl]], and [[dzaima/APL]] adopting it. However, some APLs, such as [[APL2]] and [[APLX]], keep [[Or]] as a pure [[boolean function]] and do not extend it, while [[K]] uses the | The use of GCD as an extension of [[Or]], and its extension to complex rational numbers, was proposed by [[Eugene McDonnell]] at [[APL75]].<ref>[[Eugene McDonnell]]. [https://doi.org/10.1145/800117.803810 "A Notation for the GCD and LCM Functions"] ([https://www.jsoftware.com/papers/eem/gcd.htm web]) at [[APL75]].</ref> This definition has become common among many APLs, with [[SHARP APL]], [[Dyalog APL]] (as of [[Dyalog APL 11.0|version 11.0]]), [[J]], [[NARS2000]], [[GNU APL]], [[ngn/apl]], and [[dzaima/APL]] adopting it. However, some APLs, such as [[APL2]] and [[APLX]], keep [[Or]] as a pure [[boolean function]] and do not extend it, while [[K]] uses the [[Maximum]] function as Or. | ||
== External links == | == External links == | ||
| Line 47: | Line 47: | ||
=== Documentation === | === Documentation === | ||
* [ | * [https://help.dyalog.com/17.1/#Language/Primitive%20Functions/Or%20Greatest%20Common%20Divisor.htm Dyalog] | ||
* J [https://www.jsoftware.com/help/dictionary/d101.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/plusdot#dyadic NuVoc] | * J [https://www.jsoftware.com/help/dictionary/d101.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/plusdot#dyadic NuVoc] | ||