TITLE: Suppose that $m,n \in \mathbb{Z}$ and $m$ divides $n$. Show that $\frac{\mathbb{Z}_n}{\mathbb{Z}_m} \cong \mathbb{Z}_\frac{n}{m}$
QUESTION [5 upvotes]: Suppose that $m,n \in \mathbb{Z}$ and $m$ divides $n$. Show that $$\frac{\mathbb{Z}_n}{\mathbb{Z}_m} \cong \mathbb{Z}_\frac{n}{m}$$ I try to use the third isomorphism theorem to show but I don know how to apply it here. Anyone can guide me ?

REPLY [2 votes]: You can see this problem via another point of view. For any $m$ which divides $n$ we have $$n\mathbb{Z}\lhd m\mathbb{Z}$$ and by letting $$\varphi:m\mathbb{Z}\to \mathbb{Z}/\frac nm\mathbb{Z}$$ with $$\varphi(k)=\big[{\frac1m\cdot k}\big]$$ one could check; $\varphi$ is an epimorphism and so $\ker(\varphi)=n\mathbb{Z}$. And therfore according to first isomorphism theorem, $$m\mathbb{Z}/n\mathbb{Z}\cong \mathbb{Z}/\frac nm\mathbb{Z}$$ This can lead you to have the conclusion.