What does $m\mathbb Z/n\mathbb Z$ look like?

Question

Someone asked me about what is really the quotient $m\mathbb Z/n\mathbb Z$ looks like when $m|n$. Unfortunately, I did some handy calculations for him to convince him, but it didn't work. May someone help me to have a solid explanation about this group? Thank you!

Answer 1

If $m|n$ then $n\mathbb{Z}\lhd m\mathbb{Z}$. To see it's structure, look at the homomorphism $\varphi:m\mathbb{Z}\to \mathbb{Z}/\frac nm\mathbb{Z}$ defined by $\varphi(k)=\overline{\frac1m\cdot k}$ (the coset of $\frac1m\cdot k$). You can check that $\varphi$ is an epimorphism (onto) and $\ker(\varphi)=n\mathbb{Z}$. Hence, by the first isomorphism theorem, $m\mathbb{Z}/n\mathbb{Z}\cong \mathbb{Z}/\frac nm\mathbb{Z}$

Answer 2

$nZ$ is a normal subgroup of $mZ$. Thus, $mZ/nZ$ is the factor group.