I've been given the following problem as homework:
If $\mathbb(m, n)\neq 1$, prove that $\mathbb{Z}_{mn}$ is not isomorphic to $\mathbb{Z}_{m}\times\mathbb{Z}_{n}$ (here $(m, n)$ = greatest common divisor of $m$ and $n$.)
How do I approach this problem? I think I can use the fact that there is some $c$ such that $$am + bn = c,\ c > 1,$$ but am not sure how to prove this.