To save time from writing it all out yourselves, here is what the key says. The following uses the theorem of Fundamental Theorem of Finitely Generated Abelian Groups.

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**Q1**: I have a question about the phrase "**Find all abelian groups, up to isomorphisms of order $360$**". So far the solution only lists integer groups, but it doesn't list *every* possible group in the world that are abelian, have order $360$, and are isomorphic to one (actually all) of the integer product groups listed below. Am I wrong?

**Q2**: Why is the trivial $\Bbb Z\_{360}$ not listed? It does not satisfy Fundamental Theorem of Finitely Generated Abelian Groups, I know, but the question doesn't even say I need to use it. It just says I need to find "abelian groups of order 360 + isomorphism". 

**Q3**: Relating to above, doesn't this show that $\Bbb Z\_2 \times \Bbb Z\_4 \times \Bbb Z\_3 \times \Bbb Z\_3 \times \Bbb Z\_5$ should be removed from the list since $gcd(4,2) = 2$ and therefore it the whole product can't be isomorphic to $\Bbb Z\_{360}$