I know that to define a homomorphism between cyclic groups, we only need to determine where the generator goes to. But any smart general idea to determine which choice can send zero to zero?
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If $f(1)\cdot m = 0$ in $\mathbb{Z}_n$, then $f$ can be extended to a homomorphism $f:\mathbb{Z}_m\rightarrow\mathbb{Z}_n$. Using this fact, you can count the number of homomorphisms, i.e. the number of homomorphisms is the cardinality of the set ${x\in\mathbb{Z}_n\mid x\cdot m=0}$. – Levent Dec 01 '17 at 03:28