Least common multiple and greatest common divisor.
GCD (greatest common divisor) is the largest integer that divides both. LCM (least common multiple) is the smallest positive integer both divide.
Related: LCM(a,b) × GCD(a,b) = |a × b|.
The Euclidean algorithm computes GCD efficiently by repeated division rather than checking every possible divisor, making it fast even for very large numbers — this is the same core algorithm used internally by most calculators and programming languages.
GCD(12, 18) = 6. LCM(12, 18) = 36.
Reducing fractions to simplest form, cryptography (Euclidean algorithm), and scheduling repeating events.
Adding fractions with unlike denominators; finding when two cyclical events align.
Always 1, since two distinct primes share no common factors other than 1 — this special case is called being 'coprime.'