What is the remainder when 5¹²³ is divided by 11?

Since 11 is prime and does not divide 5, Fermat's little theorem gives 5¹⁰ ≡ 1 (mod 11).

123 = 10 × 12 + 3, so 5¹²³ ≡ 5³ (mod 11).

5³ = 125 = 11 × 11 + 4, so 5³ ≡ 4 (mod 11).

Remainder = 4.

Tip: Reduce the exponent modulo (p − 1) first, then compute the small remaining power directly.