RU-ALLRUSSIAN 2023 Problem 117

We call a polynomial $P(x)$ good if the numbers $P(k)$ and $P'(k)$ are integers for all integers $k$ . Let $P(x)$ be a good polynomial of degree $d$ , and let $N_d$ be the product of all composite numbers not exceeding $d$ . Prove that the leading coefficient of the polynomial $N_d \cdot P(x)$ is integer.