A polynomial
![p(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k](/media/m/9/2/0/920ba5ee280175d0015343926aa1c895.png)
with integer coefficients is said to be divisible by an integer
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
if
![p(x)](/media/m/3/9/3/393afccb4b82415d2114a3ff957b444f.png)
is divisible by m for all integers
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
. Prove that if
![p(x)](/media/m/3/9/3/393afccb4b82415d2114a3ff957b444f.png)
is divisible by
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
, then
![k!a_0](/media/m/f/1/3/f13a97ad8d6538d3ee4d46e810a966fb.png)
is also divisible by
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
. Also prove that if
![a_0, k,m](/media/m/9/4/8/9482eb559955d26f7f493d73cd6a9d93.png)
are non-negative integers for which
![k!a_0](/media/m/f/1/3/f13a97ad8d6538d3ee4d46e810a966fb.png)
is divisible by
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
, there exists a polynomial
![p(x) = a_0x^k+\cdots+ a_k](/media/m/e/5/e/e5e8772e51c05ce2ecbc966abfeeaa20.png)
divisible by
%V0
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 non-negative integers for which $k!a_0$ is divisible by $m$, there exists a polynomial $p(x) = a_0x^k+\cdots+ a_k$ divisible by $m.$