Number of homomorphisms between finite cyclic groups ($|\mathrm{Hom}({\mathbb{Z}}_m,{\mathbb{Z}}_n)|=\gcd (m,n)$)

A group homomorphism from ${\mathbb{Z}}_m$ to the additive cyclic group ${\mathbb{Z}}_n$ is determined by the image $a$ of $1$, subject to $ma\equiv 0(\mathrm{mod}n)$. Conversely every such $a$ gives a well-defined map $k\mapsto ka$. If $d=\gcd (m,n)$, the condition is equivalent to $n/d$ dividing $a$. There are exactly $d$ such residues modulo $n$, proving the formula. More generally, homomorphisms from ${\mathbb{Z}}_m$ to any group correspond to target elements whose order of a group element divides $m$.