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gmp_hamdist gmp_gcd

Last updated: Sat, 27 Jan 2007

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# gmp_gcdext

(PHP 4 >= 4.0.4, PHP 5)

gmp_gcdext -- Calculate GCD and multipliers

## Description

array gmp_gcdext ( resource a, resource b )

Calculates g, s, and t, such that `a*s + b*t = g = gcd(a,b)`, where gcd is the greatest common divisor. Returns an array with respective elements g, s and t.

This function can be used to solve linear Diophantine equations in two variables. These are equations that allow only integer solutions and have the form: `a*x + b*y = c`. For more information, go to the "Diophantine Equation" page at MathWorld

Пример 1. Solving a linear Diophantine equation

`<?php
// Solve the equation a*s + b*t = g
// where a = 12, b = 21, g = gcd(12, 21) = 3
$a = gmp_init(12);
$b = gmp_init(21);
$g = gmp_gcd($a, $b);
$r = gmp_gcdext($a, $b);
$check_gcd = (gmp_strval($g) == gmp_strval($r['g']));
$eq_res = gmp_add(gmp_mul($a, $r['s']), gmp_mul($b, $r['t']));
$check_res = (gmp_strval($g) == gmp_strval($eq_res));
if ($check_gcd && $check_res) {
$fmt = "Solution: %d*%d + %d*%d = %d\n";
printf($fmt, gmp_strval($a), gmp_strval($r['s']), gmp_strval($b),
gmp_strval($r['t']), gmp_strval($r['g']));
} else {
echo "Error while solving the equation\n";
}
// output: Solution: 12*2 + 21*-1 = 3
?>`

add a note User Contributed Notes
gmp_gcdext
FatPhil
15-Jun-2003 06:47
`The extended GCD can be used to calculate mutual modular inverses of two
coprime numbers. Internally gmp_invert uses this extended GCD routine,
but effectively throws away one of the inverses.
If gcd(a,b)=1, then r.a+s.b=1
Therefore r.a == 1 (mod s) and s.b == 1 (mod r)
Note that one of r and s will be negative, and so you'll want to
canonicalise it.`
add a note
gmp_hamdist gmp_gcd

Last updated: Sat, 27 Jan 2007
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