Let $M$ be a free $\mathbb{Z}$-module generated by $e_{1}$ and $e_{2}$. Let $a, b$ be two non-zero integers, and $N$ be the submodule of $M$ generated by $a e_{1}+b e_{2}$. Prove that the quotient module $M / N$ is free if and only if $a, b$ are coprime.