Justify $\frac{\mathbb{Z}_a\times \mathbb{Z}_b}{\langle a/c\rangle \times \langle b/d\rangle}$ is isomorphic to $\mathbb{Z}_c\times \mathbb{Z}_d$.
Theorem: Let $M\unlhd G$ and $N\unlhd H$ as groups. Then $$(G\times H)/(M\times N)\cong (G/M)\times (H/N).$$
Proof: Let
$$\begin{align} \varphi: G\times H&\to (G/M)\times (H/N)\\ (g,h)&\mapsto (gM, hN). \end{align}$$
Clearly $\varphi$ is a well-defined, surjective homomorphism. (Why?)
Then $\ker(\varphi)=M\times N$.
Now, by the First Isomorphism Theorem,
$$(G\times H)/(M\times N)\cong (G/M)\times (H/N).\,\square$$