I need to prove/disprove whether two groups are isomorphic or not :
The two groups are : $Z_{18} \times Z_{15} \times Z_{12} $ and $Z_{90} \times Z_{36}$
Now using the property that $Z_{m} \times Z_{n}$ is isomorphic to $Z_{mn}$ iff $m $ and $n$ are co prime the first group can be transformed to
$Z_{90} \times Z_{3} \times Z_{12}$
Now using the Fundamental theorem of Finite Abelian Groups : states that" a finite Abelian group is isomorphic to a direct product of cyclic groups of prime-power order, where the decomposition is unique up to the order in which the factors are written."
Now Clearly by any means I cannot Further decompose/combine different groups to give $Z_{3} \times Z_{12}$ as $Z_{36}$
So by using above theorem two groups must be non -isomorphic.
Is my reasoning and answer correct ?