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.


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