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

Parameters

a: It can be either a GMP number resource, or a numeric string given that it is possible to convert the latter to a number.
b: It can be either a GMP number resource, or a numeric string given that it is possible to convert the latter to a number.

Return Values

An array of GMP numbers.

Examples

Example #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

?>