Question 52
Let $\mathbb{N}, \mathbb{Z}, \mathbb{Q}$ denote the set of all positive integers, integers and rational numbers respectively. Determine all functions $f : \mathbb{Q} \to \mathbb{Z}$ 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 $b \in \mathbb{N}$.
$$
f\left(\frac{f(x)+a}{b}\right) = f\left(\frac{x+a}{b}\right)
$$
for all $x \in \mathbb{Q}, a \in \mathbb{Z}$, and $b \in \mathbb{N}$.