Is $\Bbb Z/30\Bbb Z $ isomorphic to $\Bbb Z/2\Bbb Z \oplus \Bbb Z/3\Bbb Z \oplus \Bbb Z/5\Bbb Z $? Is $\Bbb Z/30\Bbb Z $ isomorphic to $\Bbb Z/2\Bbb Z \oplus \Bbb Z/3\Bbb Z \oplus \Bbb Z/5\Bbb Z $?
my reasoning for thinking this isn't the case is that the first group has an element of order 6 whereas the second doesn't.

Yes: the element $(1,1,1)$ of the latter group has order $30$, so the latter group is a cyclic group of order $30$.
