Question 24
(GBR 1) The polynomial $P(x)=a_0x^k+a_1x^{k-1}+\cdots+a_k$, where $a_0,\ldots,a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0\cdot k!$ is a multiple of $m$. Also prove that if $a,k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$ can be found that is divisible by $m$.