DE-MO 2014 Problem 2

For a positive integer $n$ , let $y_n$ be the number of $n$ -digit positive integers containing only the digits $2,3,5, 7$ and which do not have a $5$ directly to the right of a $2.$ If $r\geq 1$ and $m\geq 2$ are integers, prove that $y_{m-1}$ divides $y_{rm-1}.$