ISL 2013 N6

Determine all functions $f: \mathbb{Q} \longrightarrow \mathbb{Z}$ satisfying $$f\left(\frac{f(x)+a}{b}\right)=f\left(\frac{x+a}{b}\right)$$ for all $x \in \mathbb{Q}, a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.) (Israel)