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
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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