IR-TST 2004 Problem 6

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k$ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$