Rings and projection of a map

Question

I came across this question as I was reading and have no idea how to prove it:

Suppose we have the ring $R=\mathbb{Z}×\mathbb{Z}_3×\mathbb{Z}_5$ Consider the projection map p:R→Z_5 given by p(a,b,c)=c: - show its a subjective ring homomorphisism - Is it unital?

Answer 1

Hint show that $p(a+a‘,b+b‘,c+c‘) = p(a,b,c) + p(a‘,b‘,c‘)$, $p(a\cdot a‘,b\cdot b‘,c\cdot c‘) = p(a,b,c) \cdot p(a‘,b‘,c‘)$ and $p(1,1,1) =1$, the latter being tautological.

Hint given some $c\in \mathbb Z_5$ can you find an element $(a,b,c)\in \mathbb Z \times \mathbb Z_3 \times \mathbb Z_5$?