MathNet 2024-part-02 — 168 problems (EN–VI) Question 162

Let $n$ be a positive integer. We say that a polynomial $P$ with integer coefficients is $n$-good if there exists a polynomial $Q$ of degree 2 with integer coefficients such that $Q(k)(P(k)+Q(k))$ is never divisible by $n$ for any integer $k$.
Determine all integers $n$ such that every polynomial with integer coefficients is an $n$-good polynomial.