Question 150
As usual, let $\mathbb{Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients.
Find all functions $\theta : \mathbb{Z}[x] \to \mathbb{Z}$ 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 $\theta(p \cdot q)$.
Find all functions $\theta : \mathbb{Z}[x] \to \mathbb{Z}$ 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 $\theta(p \cdot q)$.