TSTST 2018 Problem 1

As usual, let $\mathbb{Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta: \mathbb{Z}[x] \rightarrow \mathbb{Z}$ such that for any polynomials $p, q \in$ $\mathbb{Z}[x]$, - $\theta(p+1)=\theta(p)+1$, and - if $\theta(p) \neq 0$ then $\theta(p)$ divides $\theta(p \cdot q)$.