Fairly easy. Set $x_2,\dots,x_{2016}$ such that $x_i\equiv 0\pmod{a_1}$ and fix $S:=\textstyle \sum_{2\le j\le 2016}a_j x_i^j$. We have $S\equiv 0\pmod{a_1}$. Now choose a target $T\equiv 0\pmod{a_1}$ such that $T\gg S$. Then, $T^{2017}-S\equiv 0\pmod{a_1}$, so it suffices to choose $x_1=(T^{2017}-S)/a_1$.