Linear Diophantine Equation

An equation in which both the coefficients and variables are integers, is called a Diophantine equation, named after the Greek mathematician Diophantus of Alexandria. A Diophantine equation is called linear if each term in the equation is a constant or a constant times a single variable.

Linear Diophantine Equation Theorem, Part 1 (LDET 1)

For all integers $a$, $b$ and $c$, with $a$ and $b$ both not zero, the linear Diophantine equation $ax+by=c$ (in variables $x$ and $y$) has an integer solution if and only if $d|c$, where $d=\gcd (a,b)$. → This comes from Bezout’s Lemma, see Greatest Common Divisor

Linear Diophantine Equation Theorem, Part 2, (LDET 2)

Let $a$, $b$ and $c$ be integers with $a$ and $b$ both not zero, and define $d=\gcd (a,b)$. If $x=x_0$ and $y=y_0$ is one particular integer solution to the linear Diophantine equation $ax+by=c$, then the set of all solutions is given by

$\{(x,y):x=x_0+\frac{b}{d}n,y=y_0-\frac{a}{d}n,n\in \mathbb{Z}\}$