Theorem

For any two integers $a,b\in\mathbb{Z}$ with $b\not=0$, $\exists x_{0},y_{0}\in\mathbb{Z}$ such that for $d=(a,b)$, we have by Theorem 1.1 $$ax_{0}+by_{0}=d$$Given this information we have that all integer solutions of $ax+by=d$ are given by the parameterization $$$$\begin{align*} x&=x_{0}+\frac{tb}{d}\\\\ y&=y_{0}-\frac{ta}{d} \end{align*}$$$$ for $t\in\mathbb{Z}$.