MathNet 2010 — Part 02 Question 94

Let $P(x)$ and $Q(x)$ be polynomials with integer coefficients. Let $a_n = n! + n$. Show that if $P(a_n)/Q(a_n)$ is an integer for every $n$, then $P(n)/Q(n)$ is an integer for every integer $n$ such that $Q(n) \neq 0$.