Easy# Gauss's Lemma on Primitive Polynomials with Integer Coefficients

ABSALG-EUVRK2

Let $P\in{\mathbb Z}[x]$ be a non-zero polynomial.

We say that $P$ is *primitive* when the greatest common positive divisor of the coefficients of $P$ equals $1$.

Let:

$P(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_0,\qquad Q(x)=b_mx^m+b_{m-1}x^{m-1}+\ldots+b_0$

be non-zero **primitive** polynomials in ${\mathbb Z}[x]$.

Let $p\ge 2$ be a prime number.

Which of the following is *always* true?