Let $F_n$ be the Fibonacci sequence ($F_1=1, F_2=1, F_3=2, \ldots F_n$ such that $F_n = F_{n-1} + F_{n-2}$). Show that $F_n$ divides $F_{rn}$ for all $r,n \ge 1$.
I'm not sure how to show this rigorously. I feel that eventually the terms repeat and so $F_{nr} = qF_n$, where $q$ is an integer.
(I attached the problem in an image since I used LaTex with my intuition.) I thought about using induction but got nowhere and am completely stuck. How would I start and what would I need to show? This is for an introductory number theory class.