lcms

LCMs, short for least common multiples, are a fundamental concept in number theory. The least common multiple of a set of integers is the smallest positive integer that is a multiple of each of the given numbers. For two positive integers a and b, lcm(a,b) can be computed as lcm(a,b) = |a × b| / gcd(a,b), where gcd is the greatest common divisor. This relationship shows how the LCM is connected to the GCD.

For more than two numbers, the LCM can be found recursively: lcm(a1, a2, ..., an) = lcm(lcm(a1, ..., a_{n-1}), an). The LCM is commutative and associative, meaning the order of the numbers does not affect the result, and grouping does not change the outcome.

Prime factorization provides another view: the LCM contains, for each prime, the highest power that appears in any of the factorizations. This makes the LCM particularly useful in problems involving fractions, as it enables the addition or subtraction of fractions with different denominators by converting to a common denominator.

Notes on zeros: if any input is zero, the LCM is typically defined as zero, since zero is a multiple of every integer.

Applications of the LCM include adding fractions, scheduling events with different periods, and solving problems that require a common time or cycle that aligns multiple processes. The LCM is closely related to the GCD, with the fundamental identity lcm(a,b) × gcd(a,b) = |a × b| for two integers.