Find a solution of $x^{101} ≡ 5 \pmod {13}$.
There are two cases to consider: 13 divides $x$ and 13 does not divide $x$. In the first case, there is no solution because then $x \equiv 0 \pmod {13} \implies x^{101} \equiv 0 \pmod {13}$.
So we are left with the second case where 13 does not divide $x$. This means we can use the FlT, and so we have
$x^{12} \equiv 1 \pmod {13}$
$(x^{12})^{8} \equiv x^{96} \equiv 1 \pmod {13}$
And so
$x^{101} \equiv x^{96} \times x^{5} \equiv x^5 \equiv 5 \pmod {13}$
The following table shows some $x^5 \pmod{13}$.
| x | x^5 | x^5 mod 13 |
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 2 | 32 | 6 |
| 3 | 243 | 9 |
| 4 | 1024 | 10 |
| 5 | 3125 | 5 |
| 6 | 7776 | 2 |
We can see $x=5$ is a solution, and our assumption 13 does not divide $x=5$ holds.