What is the remainder when 11¹²² is divided by 7?

11 ≡ 4 (mod 7). Check the cycle of powers of 4 mod 7: 4¹≡4, 4²≡2, 4³≡8≡1. So the order of 4 mod 7 is 3 (this divides 6, consistent with Fermat's little theorem since 7 is prime).

122 = 3×40 + 2, so 4¹²² ≡ 4² ≡ 2 (mod 7).

Tip: reduce the base mod p first, then find the actual cycle length of its powers (it may be a proper divisor of p − 1) before reducing the exponent.