Question 16
(GBR 3) A polynomial $p(x)=a_0x^k+a_1x^{k-1}+\cdots+a_k$ 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 $p(x)=a_0x^k+\cdots+a_k$ divisible by $m$.