TITLE: If we know $\mathbb{Z}_{n}/ \mathbb{Z}_{m} \cong \mathbb{Z}_{k}$, can we conclude that $\mathbb{Z}_{n} \cong \mathbb{Z}_{m} \times \mathbb{Z}_{k} $?
QUESTION [5 upvotes]: If we know $\mathbb{Z}_{n}/ \mathbb{Z}_{m} \cong \mathbb{Z}_{k}$, can we conclude that $\mathbb{Z}_{n} \cong \mathbb{Z}_{m} \times \mathbb{Z}_{k} $?

I think not, but I can not find right counterexample. Any hint helps!

REPLY [7 votes]: Counterexample: $n =4, k=2 $.
Notably, $\Bbb Z_m \times \Bbb Z_k$ is cyclic if and only if $m, k$ are relatively prime.