Determine the least non-negative remainder when $3^{2013}$ is divided by 43.
We want to solve
$$ 3^{2013} \equiv a \pmod {43} $$
Since 43 is prime and does not divide 3, we can use the FlT,
$ 3^{42} \equiv 1 \pmod {43} $
And so
$ 3^{2013} \equiv (3^{42})^{47} \times 3^{39} \equiv 3^{39} \pmod {43} $
We also use $3^{12} \equiv 4 \pmod {43}$
$ 3^{2013} \equiv (3^{12})^3 \times 3^3 \equiv 4^3 \times 3^3 \equiv 8 \pmod {43} $
So the remainder is 8 when $3^{2013}$ is divided by 43.