Let $p$ be a prime and $Q(x)$ be a polynomial with integer coefficients such that $Q(0) = 0, \ Q(1) = 1$ and the remainder of $Q(n)$ is either $0$ or $1$ when divided by $p$, for every $n \in \mathbb{N}$. Prove that $Q(x)$ is of degree at least $p - 1$.