(i) Determine the remainder when $6^{2014}$ is divided by 11.
(ii) Determine the remainder when $6^{2013}$ is divided by 11.
(i) We need to solve
$$ 6^{2014} \equiv x \pmod {11} $$
Since 11 is prime, and does not divide 6, we have by FlT
$ 6^{10} \equiv 1 \pmod {11} $
And so
$ 6^{2014} \equiv (6^{10})^{201} \times 6^4 \equiv 6^4 \equiv 1296 \pmod {11} $
$ 6^{2014} \equiv 9 \pmod {11} $
So the remainder when $6^{2014}$ is divided by 11 is 9.
(ii) This is very similar to the above. The difference leads to
$ 6^{2013} \equiv (6^{10})^{201} \times 6^3 \equiv 6^3 \equiv 216 \pmod {11} $
$ 6^{2013} \equiv 7 \pmod {11} $
So the remainder when $6^{2013}$ is divided by 11 is 7.