Linear Diophantine equation

Summary

A linear Diophantine equation in two unknowns is an equation $a x + b y = c$ with given integers $a$, $b$ and $c$. Its solutions are the pairs of integers $(x, y)$ that satisfy it. When $a$ and $b$ are not both $0$, the equation has an integer solution if and only if $\gcd(a, b)$ divides $c$. In this case, let $d = \gcd(a, b)$ and let $(x_0, y_0)$ be one solution. Then the solutions are the pairs $(x_0 + k b / d, \; y_0 - k a / d)$ with $k$ an integer.