$\frac{2016^{2016}-3}{3}=2016^{2015}\cdot 672-1$. Checking $2,3,5,7$ we see easily that all of them don't work. Checking $11$, we get $2016^{2015}\cdot 672-1\equiv 3^{2015}-1\equiv 0 \pmod {11}$ so the answer is $11$.
Let's be a bit more thorough. $2,3,7$ don't work because they are factors of $2016$.
$5$ doesn't work:
$672\times 2016^{2015}-1\equiv (2\times 1^{2015}-1)\mod 5\equiv 1\mod 5$.
$11$ works:
$672\times 2016^{2015}-1\equiv (1\times 3^{2015}-1)\mod 11\equiv (1\times 1^{403}-1)\mod 11\equiv 0\mod 11$.