# gmp_gcdext

(PHP 4 >= 4.0.4, PHP 5, PHP 7, PHP 8)

gmp_gcdextCalculate GCD and multipliers

### Description

gmp_gcdext( `\$num1`, `\$num2`): array

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

`num1`

A GMP object, an int or a numeric string.

`num2`

A GMP object, an int or a numeric string.

### 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?>`

`The extended GCD can be used to calculate mutual modular inverses of twocoprime 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=1Therefore 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 tocanonicalise it.`