Solving Congruences

We wish to solve equations of the form $a x \equiv b \pmod{m}$ given $a,b \in \mathbb{Z}$ and $m \in \mathbb{N}$ for $x \in \mathbb{Z}$. We can often simplify these equations, for instance $7 x \equiv 3 \pmod{5}$ reduces to $x \equiv 4 \pmod{5}$ (since $21 \equiv 1$ and $9 \equiv 4 \pmod{5}$). This equations are not always soluble, for instance $6 x \equiv 4 \pmod{9}$, as $9 \nmid 6 x - 4$ for any $x \in \mathbb{Z}$.