IN-TST 2001 Problem 5

Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ , $2$ , $3$ , $4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$ . Prove that $p$ divides all the coefficients of $Q(x)$ .