Morphisms between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$

Question

I'm trying to determinate how many morphisms of groupes exist between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$ for $n,m\in\mathbb{N}$.

I know a morphism is determinated by the image of the class $1$ and all image $g$ of $1$ have to satisfy $gn=0$.

My question is, how many elements $g$ like that exist and why?

Answer 1

Hint Let $1\in \Bbb Z/n\Bbb Z$ be your generator, $\eta$ a morphism. Then the order of $\eta(1)\in \Bbb Z/m \Bbb Z$ divides the order $n$ of $1$ in $\Bbb Z/n\Bbb Z$ and $m$, so ${\rm ord}\,\eta(1)\mid (n,m)$.

Answer 2

Hint $\,\ nk\equiv 0\pmod m\!\iff\! m\mid nk\!\iff\! m\mid (nk,mk)=(n,m)k\!\iff\! m/(n,m)\mid k$