What is $\mbox{Hom}(\mathbb{Z},\mathbb{Z}/n\mathbb{Z})$?

Question

What is $\mbox{Hom}(\mathbb{Z},\mathbb{Z}/n\mathbb{Z})$? Will it be cyclic always?

I cannot find it explicitly but I understand that there are $d(n)(=$ Number of divisors of $n)$ number of elements.I need it in modulo $\mathbb{Z}$ version.

Help me ,thanks.

Answer 1

For any commutative group $G$, you have an isomorphism $\,\,\varphi\colon \operatorname{Hom}(\mathbf Z, G) \simeq G$, defined by $\varphi(f)=f(1)$.