gmp_gcdext

Description

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 ?>