$ p(x)$ is a polynomial in $ \mathbb{Z}[x]$ such that for each $ m,n\in \mathbb{N}$ there is an integer $ a$ such that $ n\mid p(a^m)$. Prove that $0$ or $1$ is a root of $ p(x)$.
Problem
Source: Iranian National Olympiad (3rd Round) 2004
Tags: algebra, polynomial, number theory proposed, number theory