Question 103
Let $\mathbb{Z}[X]$ be the set of polynomials with integer coefficients. Find all functions $f: \mathbb{Z}[X] \rightarrow \mathbb{Z}[X]$ such that for all $P, Q \in \mathbb{Z}[X]$ and $r \in \mathbb{Z}$, we have
$$
P(r)|Q(r) \Longleftrightarrow (f(P))(r)|(f(Q))(r)
$$
$$
P(r)|Q(r) \Longleftrightarrow (f(P))(r)|(f(Q))(r)
$$