Number Theory

Prime factorization, the Euclidean algorithm, and counting divisors.

Congruences, modular inverses, and working with remainders.

Fermat's Little Theorem, Euler's theorem, and multiplicative order.

Solving systems of congruences with coprime moduli.

Integer solutions to ax + by = c.

Pythagorean triples and sums of two squares.

Legendre's formula and p-adic valuation for factorials.

Counting and summing the divisors of an integer.

Divisibility of binomial coefficients and Lucas's theorem.

Base conversion, digit sums, and divisibility tests.

Using quantities that never change to prove something is impossible.

Applying the pigeonhole principle to remainders and subset sums.