Finding two words of lengths that are relatively prime in a regular language?
Consider a minimal DFA for $L$. Each state in the DFA has one outgoing edge, so the DFA looks like a path entering a cycle. That means that $W = \{ n : a^n \in L \}$ is eventually periodic: for some $k,l$, for all $n \geq k$ it holds that $n \in W$ iff $n + l \in W$. We are now left with a problem in number theory, which I leave for you to ponder. (Try a few examples to see when an eventually periodic $W$ can fail to satisfy the property you're looking for.)