Calculate greatest common divisor of numbers.
Understanding greatest common divisor and number theory. Greatest common divisor (GCD) represents largest number dividing two or more numbers without remainder. Example: GCD(12, 18) = 6 (6 divides both 12 and 18). GCD(20, 30, 40) = 10 (10 divides all three). Euclidean algorithm: efficient method computing GCD—repeatedly divide larger by smaller, replacing larger with remainder, until remainder zero. Prime factorization method: break numbers into prime factors, GCD equals product of common factors at lowest powers.
Example: 24 = 2³ × 3, 36 = 2² × 3². Common factors: 2² × 3 = 12. Applications: fraction simplification (GCD determines lowest common denominator), scheduling (finding repeating cycles), distribution (dividing items into equal groups). Fractions: reducing 18/24 to 3/4 by dividing by GCD(18,24) = 6. Clock problems: hands align periodically based on GCD calculations. Cutting problems: maximum piece size from materials determined by GCD.
Relationship with LCM (least common multiple): GCD × LCM = product of two numbers. Relatively prime numbers: GCD = 1 (numbers share no common factors). Cryptography: GCD used in encryption algorithms, public-key systems. Number theory foundations: GCD essential for modular arithmetic, number patterns. Computing GCD: calculators provide instant results, understanding underlying concepts enables verification. Applications spanning arithmetic, algebra, geometry, cryptography, computer science. Understanding GCD enables fraction simplification, optimization problems solving, mathematical pattern recognition..