Proving that order of a cyclic group $G$ is divisible by a number

Question

Suppose that $\exists x,s.t. |x|=20$ and $\exists y,s.t. |y|=16$ and $x,y \in G$ where $G$ is a cyclic group.

How can I show that order of $G$ is divisible by 80?

I was thinking of using the fact that "$G$ is a finite cyclic group and $b\in G$ means that $|b|$ divides order of $G$. But not sure how to apply it in this context.. Thanks!

Answer 1

Since $5|20|n=|G|$ and $16|n$ and $(5,16)=1$, $5\cdot 16|n$.

Answer 2

Hint. A cyclic group is abelian. What can you say about the order of $xy$?