Showing the existence of an element $g \in G$, a finite abelian group, where $g$ has order $LCM\{|x|: x \in G\}$

Question

So I, more or less, need help understanding this problem.

The first thing I have tried is to pick a specific group: $\mathbf{Z}_5$, under multiplication.

To demonstrate, say we pick element $2$ and $|2| = 4$, and then the only other possible orders are $\{1,4\}$.

Indeed, $4 = LCM(1,4)$.

Not really quite sure, how to generalize this, yet. Any hints? What shall I think about? Definitely prime factorization of each element, hmm... I think!