Let $a,b\geq 2$ be integers and $N$ the image of the homomorphism $$\mathbb{Z}_a\rightarrow \mathbb{Z}_{ab}, \;\;\;[k]_a\mapsto[kb]_{ab}$$ Identify the quotient group $\mathbb{Z}_{ab}/N$ with a familiar group.
I have done quite a bit of thinking about this problem, and it seems that the best strategy would be to try and find an isomorphism from $\mathbb{Z}_{ab}/N$ to another group such that the kernel is $N$. The other group should be a familiar group. If this is the correct approach, I'm not sure how to proceed. If it isn't, I would greatly appreciate any hints/guidance.
Let $\varphi$ be the homomorphism. We have that $$\Bbb{Z}_a/\ker \varphi \simeq N.$$ Take some $k \in \Bbb{Z}_a$, then $[k]_a \in \ker\varphi$ iff $[kb]_{ab} = [0]_{ab}$.
Now, finding $\ker\varphi$ is easy to get $N$ and so $\Bbb{Z}_{ab}/N$.