# How many surjective homomorphisms are there from ZZ_10 onto ZZ_5 ?

There are $4$
The (group) homomorphisms from ${\mathbb{Z}}_{10}$ onto ${\mathbb{Z}}_{5}$ are determined by the value to which $\hat{1}$ is mapped (since $\hat{1}$ generates ${\mathbb{Z}}_{10}$). Any of the $4$ non-zero members of ${\mathbb{Z}}_{5}$ are possible, since they are all of order $5$ and generate ${\mathbb{Z}}_{5}$.
So there are $4$ possible surjective homomorphisms.