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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

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
?>

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gmp_gcdext

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