What is the remainder when $7^{77}$ is divided by 13?

By Fermat's Little Theorem: $7^{12} \equiv 1 \pmod{13}$. $77 = 12 \times 6 + 5$, so $7^{77} \equiv 7^{5} \pmod{13}$. $7^{5} = 16807 = 13 \times 1292 + 11$, so the remainder is 11.