What is the remainder when 8²¹⁵ is divided by 13?

Since 13 is prime and does not divide 8, Fermat's little theorem gives 8¹² ≡ 1 (mod 13).

215 = 12 × 17 + 11, so 8²¹⁵ ≡ 8¹¹ (mod 13).

8² = 64 ≡ 12 ≡ −1 (mod 13), so 8¹¹ = 8¹⁰ × 8 ≡ (8²)⁵ × 8 ≡ (−1)⁵ × 8 = −8 ≡ 5 (mod 13).

Tip: Finding a power congruent to ±1 (here 8² ≡ −1) shortens the exponent arithmetic dramatically.