LCM

The least common multiple (LCM) of a set of integers is the smallest positive integer that is a multiple of each number in the set. For two integers a and b, the LCM is the least positive number divisible by both a and b. The operation is commutative and associative, so the LCM of a set can be computed by repeatedly taking the LCM of pairs.

A common way to compute lcm(a,b) is via the greatest common divisor: lcm(a,b) = |a × b| / gcd(a,b) for nonzero a and b. Another method uses prime factorization: if a = ∏ p_i^{α_i} and b = ∏ p_i^{β_i}, then lcm(a,b) = ∏ p_i^{max(α_i, β_i)}. For more than two numbers, lcm(a1, a2, ..., an) can be found by iteratively taking pairwise LCMs, or by taking the product of primes raised to the maximum exponent that appears in any factorization.

Several properties are noteworthy. The LCM is related to the greatest common divisor via gcd(a,b) × lcm(a,b) = |a × b| for integers a and b. LCMs are used to find common denominators when adding or comparing fractions and to synchronize cycles or events that must coincide. Special cases include lcm(0, n) = 0 for n ≠ 0 by common convention, while lcm(0,0) is undefined or defined as 0 in some contexts.