Confusion with quotient module of cyclic groups

Question

Take $\mathbb Z/ m \mathbb Z$, $n(\mathbb Z/ m \mathbb Z)$, then we can consider their quotient. Since $n(\mathbb Z/ m \mathbb Z) = n \mathbb Z/m \mathbb Z$, then by the third isomorphism theorem of modules we know the quotient should be $\mathbb Z/n \mathbb Z$. This is wrong judging from the fact that $Ext_1^{\mathbb Z}(\mathbb Z/ m \mathbb Z, \mathbb Z/ n \mathbb Z) = (\mathbb Z/ n\mathbb Z)/m(\mathbb Z/ \mathbb Z)=\mathbb Z/ gcd(m,n)\mathbb Z$. Why is it wrong?

Answer 1

You can only apply the third isomorphism theorem in your example if $m\Bbb{Z}$ is a subgroup of $n\Bbb{Z}$, i.e, if $n$ divides $m$.