Putnam 2016 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.