It occurs to me the following is true: $$(\mathbb{Z}_n \times \mathbb{Z}_m) / \mathbb{Z}_k \cong \mathbb{Z}_{n/k} \times \mathbb{Z}_m$$ when $k \mid n$. But I fail to see the way to prove that.
The way I want to use it is to show $$(\mathbb{Z}_{10} \times \mathbb{Z}_{20}) / \mathbb{Z}_2 \cong \mathbb{Z}_5 \times \mathbb{Z}_{20}$$ My idea is to start by decomposing the direct product as following: $$(\mathbb{Z}_{10} \times \mathbb{Z}_{20}) / \mathbb{Z}_2 \cong (\mathbb{Z}_2 \times \mathbb{Z}_5 \times \mathbb{Z}_{20}) / \mathbb{Z}_2 \cong \mathbb{Z}_5 \times \mathbb{Z}_{20}$$ However, I am still unsure whether the last step is always true, and I would really appreciate some help!
Summarizing the comments; it is not true in general. The simplest example is the quotient $$(\Bbb{Z}_4\times\Bbb{Z}_2)/\Bbb{Z}_2.$$ Then the isomorphism type of the quotient depends on whether you take the quotient w.r.t. the subgroup $\Bbb{Z}_2\subset\Bbb{Z}_4$ of the first factor, or the subgroup $\Bbb{Z}_2\subset\Bbb{Z}_2$ of the second factor. And there is even a third subgroup $\Bbb{Z}_2\subset\Bbb{Z}_4\times\Bbb{Z}_2$, that is not contained in either factor.