Was my abstract algebra textbook trying to kill me?
I assume that Sylow theorems were not covered yet?
And since the theorem holds also for non-abelian groups, I wonder why they restrict to abelian groups. It makes the following easier though:
Let $G$ be a group of order $n$ divisible by $3$. Select $h\in G\setminus\{1\}$. If the order of $h$ is divisible by $3$, then you find a power of $h$ that has order $3$. Otherwise $G/\langle h\rangle$ has order divisible by $3$ and can be assumed by induction to have an element $g+\langle h\rangle$ of order $3$, which has order a multiple of $3$ in $G$.