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Eulerโs Theorem says that if a and n are coprime, then a raised to the power ฯ(n) is congruent to 1 modulo n.
Fermat's Little Theorem says that for a prime p and integer a not divisible by p, a^{p-1} โก 1 (mod p).
Fast exponentiation (binary exponentiation) computes a^n using repeated squaring in O(log n) multiplications.
The Chinese Remainder Theorem (CRT) reconstructs an integer from its remainders modulo pairwise coprime moduli and guarantees a unique answer modulo the product.
The linear sieve builds all primes up to n in O(n) time by ensuring each composite is marked exactly once by its smallest prime factor (SPF).
Prime factorization expresses any integer greater than 1 as a product of primes raised to powers, uniquely up to ordering.
The Extended Euclidean Algorithm finds integers x and y such that ax + by = gcd(a, b) while also computing gcd(a, b).
The Sieve of Eratosthenes marks multiples of each prime to find all primes up to n in O(n log log n) time.
The greatest common divisor (gcd) of two integers is the largest integer that divides both without a remainder.
Randomized algorithms use coin flips (random bits) to guide choices, often making code simpler and fast on average.
Matrix exponentiation turns repeated linear transitions into a single fast power of a matrix using exponentiation by squaring.
Small-to-large merging is a technique where you always merge the smaller container into the larger one to guarantee low total work.