Given that the order of 5 modulo 13 is 4, determine the least non-negative residue $x$ such that $5^{101} \equiv x \pmod {13}$.
We're given
$$ 5^4 \equiv 1 \pmod 13 $$
Using $101 = (25 \times 4) + 1$, we have
$$ 5^{101} \equiv (5^4)^25 \times 5^1 \equiv (1) \times 5 \equiv 5 \pmod {13} $$
And so the least non-negative residue $x$ is 5.