= Number of homomorphisms between finite cyclic groups
{title2=$|\operatorname{Hom}(\mathbb Z_m,\mathbb Z_n)|=\gcd(m,n)$}
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 $ma\equiv0\pmod n$. 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 $m$.
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