Find the least nonnegative residue Find the least nonnegative residue of $5^{18} \mod 11$. 
To do this I took $5^2 \equiv 3 \mod 11$. Then I did $(5^2)^5 \equiv 3^5 \mod 11$. And $3^5 \equiv 1 \mod 11$. 
So now I have $5^{10} \equiv 1 \mod 11$. Then I multiplied both sides by $5^8$ to get $5^{18} \equiv 5^8 \mod 11$.  So I believe $5^8$ is the least nonnegative residue but I am not entirely sure. Can someone please confirm that this is correct?

As you mentioned, we have $5^2 \equiv 3$ mod 11. However, I find it easier to proceed by showing,
$$5^8 \equiv (5^2)^4 \equiv 3^4 \equiv 4 \text{ mod }11, $$
so that $$5^{16} \equiv (5^8)^2 \equiv 4^2 \equiv 5 \text{ mod }11,$$ and then finally,
$$ 5^{18} \equiv 5^2\cdot5^{16} \equiv 3 \cdot 5 \equiv 4 \text{ mod } 11.$$
Also, this should be a much more satisfying answer since the least nonnegative residue of $n$ will always be in the interval $[0,n]$.
