GCF and LCM Calculator - Greatest Common Factor and LCM

The Greatest Common Factor (GCF) and the Least Common Multiple (LCM) are two of the most fundamental concepts in number theory. The GCF (also called the Greatest Common Divisor or GCD) of a set of integers is the largest positive integer that divides each of them without a remainder. The LCM is the smallest positive integer that is divisible by each number in the set. Together, they appear in countless mathematical and practical applications, from simplifying fractions to scheduling and engineering problems. The most efficient algorithm for computing the GCF of two numbers is the Euclidean algorithm, discovered in ancient Greece and still used today. It works by repeatedly replacing the larger number with the remainder when it is divided by the smaller, until the remainder reaches zero. The last nonzero remainder is the GCF. For example, GCF(48, 18): 48 = 2 * 18 + 12, then 18 = 1 * 12 + 6, then 12 = 2 * 6 + 0, so GCF = 6. Once the GCF is known, the LCM can be computed using the identity LCM(a, b) = |a * b| / GCF(a, b). This avoids listing all multiples and works efficiently even for large numbers. For more than two numbers, the GCF and LCM are computed iteratively: GCF(a, b, c) = GCF(GCF(a, b), c), and similarly for LCM. In everyday life, the GCF is used to simplify fractions: the fraction a/b is in lowest terms when GCF(a, b) = 1. The LCM is used when adding or subtracting fractions with different denominators — the common denominator is the LCM of the original denominators. In scheduling, the LCM tells you when two recurring events will coincide. For example, if one event repeats every 4 days and another every 6 days, they align every LCM(4, 6) = 12 days. This calculator supports any number of positive integers and uses an efficient iterative Euclidean algorithm. Results are computed instantly in your browser with no data sent to any server.