Proof regarding cyclic groups

Question

I need to prove the following: G is cyclic iff for every m s.t m divides |G|, G has an element of order m.

I'm able to convince myself that its true with examples but not sure how to go about proving it formally. Can someone help me with this?

Answer 1

It follows from those assumptions that $G$ has an element $g$ of order $\lvert G\rvert$. But then $\langle g\rangle=G$.