# $gcd(a_0,a_1,…,a_n,m)=1$ and For every integer $k$, $m|P(k)$

Given a positive integer $$n$$. How many positive integers $$m$$ are there so that there exists a polynomial $$P(x)=a_nx^n+...+a_1x+a_0$$ with integer coefficients and $$deg(P(x))=n$$ satisfying.

1. $$gcd(a_0,a_1,...,a_n,m)=1$$

2. For every integer $$k$$, $$m|P(k)$$

If $$m \leq n$$, then I think $$P(x)=(x-1)(x-2)...(x-n)$$ satisfy the condition, but I cannot prove or disprove it. How can I solve this problem, with $$m \leq n$$ and $$m >n$$ ?

Sorry for my grammar, English is my second language.