I was working on: https://math.stackexchange.com/questions/161578/kind-of-functional-eq-in-integers
I found a sort of way...
but I need to show that:
$p(x) \mid q(x)$ for infinite values of $x$ (integer) implies $p(n) \mid q(n) \quad \forall n \in \mathbb{Z}$
I was told that it is always true, but I can show it just for polynomials. Is it a well known theorem, or is there an easy proof?