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Drexel28 In fact, $\displaystyle \mathbb{Z}_n\oplus\mathbb{Z}_m$ is cyclic iff $\displaystyle (m,n)=1$. To think of why this is we need some element of $\displaystyle \mathbb{Z}\oplus\mathbb{Z}_m$ to be of exactly order $\displaystyle mn$ but it is clear the order of $\displaystyle \mathbb{Z}_n\oplus\mathbb{Z}_m$ is $\displaystyle \text{lcm}(m,n)$. Therefore lastly noting that $\displaystyle \text{lcm}(m,n)=\frac{|mn|}{(m,n)}$ we conclude that $\displaystyle (m,n)=1$. And since all cyclic groups of order $\displaystyle \ell$ are isomorphic to $\displaystyle \mathbb{Z}_{\ell}$ this is the case for any cyclic groups.