CMO 2013 Problem 5

For any positive integer $n$ and $0 \leqslant i \leqslant n$ , denote $C_n^i \equiv c(n,i)\pmod{2}$ , where $c(n,i) \in \left\{ {0,1} \right\}$ . Define $$f(n,q) = \sum\limits_{i = 0}^n {c(n,i){q^i}}$$ where $m,n,q$ are positive integers and $q + 1 \ne {2^\alpha }$ for any $\alpha \in \mathbb N$ . Prove that if $f(m,q)\left| {f(n,q)} \right.$ , then $f(m,r)\left| {f(n,r)} \right.$ for any positive integer $r$ .