Exercise :
Given the following groups : $$\mathbb Z_{3^2} \times \mathbb Z_{5^2} \cong \mathbb Z_{225} $$ $$\mathbb Z_{3} \times \mathbb Z_{3} \times \mathbb Z_{5^2} \cong \mathbb Z_3 \times \mathbb Z_{75}$$ $$\mathbb Z_{3^2} \times \mathbb Z_{5} \times \mathbb Z_{5} \cong \mathbb Z_{45} \times \mathbb Z_5$$ $$\mathbb Z_{3} \times \mathbb Z_{3} \times \mathbb Z_{5} \times \mathbb Z_{5} \cong \mathbb Z_{15} \times \mathbb Z_{15}$$ show that $2$ of them are not isomorphic.
Attempt :
For the first of two that I have been asked to find, one can see that :
$$\mathbb Z_3 \times \mathbb Z_3 \times \mathbb Z_{5^2} \ncong \mathbb Z_3 \times \mathbb Z_3 \times \mathbb Z_5 \times \mathbb Z_5$$ since if this was an isomorphism, it's clear that the $\mathbb Z_3 \times \mathbb Z_3$ parts are isomorphic, which leads that it also should be $\mathbb Z_{5^2} \cong \mathbb Z_5 \times \mathbb Z_5 $. This is not possible though, as this would imply that the group $\mathbb Z_5 \times \mathbb Z_5$ is cyclic, which is false.
Similar goes for the case :
$$\mathbb Z_{3^2} \times \mathbb Z_5 \times \mathbb Z_5 \ncong \mathbb Z_3 \times \mathbb Z_3 \times \mathbb Z_5 \times \mathbb Z_5$$
Is the above approach correct and complete ? Could it be phrased better if so ?
$\mathbb Z_{225}=\mathbb Z_{15^2}$ is not isomorphic to $\mathbb Z_{15} \times \mathbb Z_{15}$ since $(15, 15)\neq1$.