2016 Putnam A1

Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer

$$

p^{(j)}(k) = \left. \frac{d^j}{dx^j} p(x) \right|_{x=k}

$$

(the $j$-th derivative of $p(x)$ at $k$) is divisible by 2016.

找到最小正整数 $j$，使得对于每个具有整数系数的多项式 $p(x)$ 以及对于每个整数 $k$，整数

$$

p^{(j)}(k) = \left。 \frac{d^j}{dx^j} p(x) \right|_{x=k}

$$

($p(x)$ 在 $k$ 处的第 $j$ 导数)可被 2016 整除。