HMMT 2015 T9

Let $z=e^{\frac{2 \pi i}{101}}$ and let $\omega=e^{\frac{2 \pi i}{10}}$. Prove that

$$\prod_{a=0}^{9} \prod_{b=0}^{100} \prod_{c=0}^{100}\left(\omega^{a}+z^{b}+z^{c}\right)$$

is an integer and find (with proof) its remainder upon division by 101.