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For centuries, number theory was considered the purest of pure mathematics — beautiful, useless, untethered from application. Then public-key cryptography arrived, and suddenly questions about prime factorization were guarding every credit card transaction on earth.

Raise a number to a huge power mod a prime — Fermat's Little Theorem tells you the answer without doing the work. It's the reason RSA encryption is computationally feasible.

17 and 5 are different numbers — but mod 12, they're identical. Congruences formalize this idea and unlock the arithmetic that powers cryptography, calendar math, and number theory.

After 12 comes 1 again — clocks have been doing modular arithmetic forever. That same wrapping logic now secures your banking app and underpins most modern cryptography.

The least common multiple is the first moment two repeating cycles land on the same beat. Whether you're scheduling tasks or reasoning about number theory, the LCM turns a timing problem into elegant arithmetic.

Two numbers share hidden structure, and the GCD is what they have in common. Euclid's algorithm finds it in milliseconds — and it underpins everything from simplifying fractions to modern cryptography.

Why does a number divisible by 9 always have digits that sum to 9? Divisibility rules aren't just shortcuts — they reveal the deep structure baked into how we write numbers.