OFM 2021 S1

Let $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ be positive integers such that $a_{n+2} = a_n + a_{n+1}$ and $b_{n+2} = b_n + b_{n+1}$ for all $n \ge 1$ . Assume that $a_n$ divides $b_n$ for infinitely many values of $n$ . Prove that there exists an integer $c$ such that $b_n = c a_n$ for all $n \ge 1$ .