Fermat's Little Theorem

What is 2¹⁰⁰ mod 5? Computing 2¹⁰⁰ directly means a number with 31 digits. But Fermat, in 1640, found a shortcut that gives the answer in one line: because 5 is prime, 2⁵ ≡ 2 (mod 5), so 2¹⁰⁰ = (2⁵)²⁰ ≡ 2²⁰ ≡ … all the way down. The theorem is one sentence, and it underpins primality tests and the encryption that protects the internet.