Original problem: what is $800^{35} \bmod 11$?
I was able to use Euler's Theorem, $a^{\varphi(n)}\equiv1\space(\bmod\space n)$, to boil the problem down to $800^{35} \equiv 800^5\space (\bmod\space 11)$.
My issue is that $800^5$ is still a pretty big number to work with (won't fit on my calculator screen) and I would like to know how to this by hand. How should I proceed from here?