Let $z,w$ be complex and equal to $e^{2\pi i/101}$ and $ e^{2\pi i/10}$, respectively.
Show that $$\prod_{a=0}^9 \prod_{b=0}^{100}\prod_{c=0}^{100}(w^a+z^b+z^c)$$ is an integer and find the remainder upon division by $101$.
I tried converting the product to a sum by logarithms but I really couldn't go any further.