The remainder when $5^{99}$ is divided by 13 is:

$5^2=25$ leaves remainder 12, i.e. −1 (mod 13), so $5^4$ ≡ 1 (mod 13) and the remainders repeat every 4 steps. Since $99=4\times 24+3$, we need $5^3=125=13\times 9+8$. The remainder is 8.