smallest positive integer $r$ so that $5^{33333}≡ r (mod 11)$ Title is pretty self explanatory. I tried using the division algorithm to get some a hint about the $5^{33333}$. This was not helpful.

First check that $5^{10} \equiv 1 \pmod{11}$. Now divide the exponent to get $33333 \equiv 3 \pmod{10}$. 
This gives $$5^{33333} \equiv 5^{3} \equiv 4 \pmod{11}.$$
