The polynomial
, where
are integers, is said to be divisible by an integer
if
is a multiple of
for every integral value of
. Show that if
is divisible by
, then
is a multiple of
. Also prove that if
are positive integers such that
is a multiple of
, then a polynomial
with leading term
can be found that is divisible by
%V0
$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ 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 $m.$
Školjka