Prove:
a) $\mathbb{Z}_{2} \times \mathbb{Z}_2 \ncong \mathbb{Z}_4$
b) $\mathbb{Z}_2 \times \mathbb{Z}_5 \cong \mathbb{Z}_{10}$
For a) I figured out that in $\mathbb{Z}_2 \times \mathbb{Z}_2$ every element is its own inverse and in $\mathbb{Z}_4$ that is not the case and I'm not sure if that is enough to show they aren't isomorphic.
For b) I don't know how to find isomorphism.
There's a theorem stating that $C_m \times C_n$ is cyclic (and then isomorphic to $C_{mn}$) if and only if $\operatorname{gcd}(m,n)=1$.