<s>[INST] 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.
 [/INST] $$5\equiv 5\mod 11\\
5^2=25\equiv 3\mod 11\\
5^3= 5^2\cdot 5\equiv 3\cdot 5=15\equiv 4\mod 11\\
5^4= 5^3\cdot 5\equiv 4\cdot 5=20\equiv 9\mod 11\\
5^5= 5^4\cdot 5\equiv 9\cdot 5=45\equiv 1\mod 11\\
$$
So, you know that $5^5\equiv 1\mod 11$.
You also know that $33333=5\cdot 6666+3$
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