IMO 1968 SL 16

Origin: GBR

Problem

A polynomial $p(x) = a_0 x^k + a_1 x^{k-1} + \cdots + a_k$ with integer coefficients is said to be divisible by an integer $m$ if $p(x)$ is divisible by $m$ for all integers $x$. Prove that if $p(x)$ is divisible by $m$, then $k! a_0$ is also divisible by $m$. Also prove that if $a_0, k, m$ are nonnegative integers for which $k! a_0$ is divisible by $m$, there exists a polynomial $p(x) = a_0 x^k + \cdots + a_k$ divisible by $m$.