TITLE: When are the product of cyclic groups also cyclic?
QUESTION [6 upvotes]: In general, products of cyclic groups are not cyclic. If $C_n$ is the cyclic group of order $n$, $C_{22} \times C_{33}$ is not cyclic. But is there an easy way to tell when such a product will be cyclic?

REPLY [3 votes]: If gcd$(m, n)=1$, we can say that $C_m\times C_n\cong C_{mn}$. The proof is simple, there must be an element $a\in C_m$ of order $m$ and an element $b\in C_n$ of order $n$.
Then, $(a,b)$ will have order $mn$, which means $C_m\times C_n$ must be cyclic.