Question 74
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 $b \in \mathbb{Z}_{\gt 0}$. (Here, $\mathbb{Z}_{\gt 0}$ denotes the set of positive integers.)
$$
f\left(\frac{f(x)+a}{b}\right)=f\left(\frac{x+a}{b}\right)
$$
for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{\gt 0}$. (Here, $\mathbb{Z}_{\gt 0}$ denotes the set of positive integers.)