Suppose you have two polynomials $P$ and $Q$ with integer coefficients. Let their GCD (more specifically, the smallest integer multiple of the GCD in $\mathbb Q$ such that all the coefficients are integral) be $D$.

If, for a particular value $x_0$, we have that $P(x_0)$ and $Q(x_0)$ are both integers, does it follow that $D(x_0)$ is an integer? Does $D(x_0)$ divide $P(x_0)$ and $Q(x_0)$?