CN-TST 2018 Problem 12

Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$ , and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions: (1) For all $n\in \mathbb{Z}$ , $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$ (2) For all $n\ge M$ , $f(n)\ge 0$ (3) For all $n>m>M$ , $n-m|f(n)-f(m)$ Show that $a$ is a perfect $r$ -th power.