Theorem

For any two integers $a,b$ with $b\not=0$ $\exists x_{0},y_{0}\in\mathbb{Z}$ such that for $d=(a,b)$ we have $$ax_{0}+by_{0}=d$$It is clear that there are infinitely many such integers because given one pair $x_{0},y_{0}$ we also have $$a(x_{0}+tb)+b(y_{0}-ta)=d$$ $\forall t\in\mathbb{Z}$.