Prove that if $f(x)=a_0x^n+a_1x^{n-1}+...+a_n$ and $m$ is an integer, then $k!|f^{(k)}(m)$ where $f^{(k)}$ is the $k^{th}$ derivative of $f$.
I do not know where to even begin this problem
2026-04-01 09:57:01.1775037421
Polynomial Congruences
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Hint: The $k$th derivative of $x^j$ is $j(j-1)\cdots(j-k+1)x^{j-k}$. If you divide the coefficient by $k!$, can you recognize the expression as a combinatorial quantity that counts something, hence is an integer?