I've been trying to find an answer for this problem. All I know at the moment is that I have to use the third isomorphism theorem. Could someone help me out?
Problem:
Let $m, n$ be positive integers such that $n | m$. Prove that $n \mathbb{Z} / m \mathbb{Z} ≃ \mathbb{Z}_{\frac{m}{n}}$.
Thank you so much in advance.
Let $m=dn$ and consider the following commutative diagram with exact rows:$\require{AMScd}$ \begin{CD} \{0\}@>>>d\Bbb Z@>>>\Bbb Z@>>>\Bbb Z/d\Bbb Z@>>>\{0\}\\ @.@V\sim VV@V\sim VV@VVV\\ \{0\}@>>>m\Bbb Z@>>>n\Bbb Z@>>>n\Bbb Z/m\Bbb Z@>>>\{0\}\\ \end{CD} where the left-handed vertical homomorphisms are the multiplication by $n$. A diagram chasing show that the right-handed vertical homomorphism is an isomorphism as well.