Q:

Endomorphisms of $\mathbb{Z}/6\mathbb{Z}$

I have a question, which may be trivial, but I really don't understand how I should solve the next exercise:

List all group endomorphisms of $\mathbb{Z}_6$. 

I consider that I must write down all the functions $f\colon \mathbb{Z}_6\to\mathbb{Z}_6$. But which is the general form of the function? 

A:

Not all functions, but only those that are additive homomorphisms. That is, only functions $f\colon\mathbb{Z}_6\to\mathbb{Z}_6$ such that $f(a+b) = f(a)+f(b)$ for all $a,b\in\mathbb{Z}_6$.

Show that if $f\colon\mathbb{Z}_6\to\mathbb{Z}_6$ is an additive homomorphism, and you know what $f(1)$ is, then you know what $f(a)$ is for all $a\in\mathbb{Z}$.
Determine what values $f(1)$ can be to give you an additive homomorphism.

