The least common multiple is the smallest positive integer that is a multiple of every number in a set. Its most common use is adding or subtracting fractions: the LCM of the denominators is the least common denominator. It also answers scheduling questions, such as when two repeating cycles next line up. Unlike the greatest common factor, which can be no larger than the smallest number in the set, the LCM is at least as large as the largest number.
Formula
For two positive integers a and b:
LCM(a, b) = (a × b) ÷ GCF(a, b)
- a, b: the two numbers.
- GCF(a, b): their greatest common factor.
This follows from the identity GCF(a, b) × LCM(a, b) = a × b. Using prime factorization instead, the LCM is the product of every prime that appears in either number, each raised to the largest exponent that appears. For three or more numbers, apply the rule repeatedly: LCM(a, b, c) = LCM(LCM(a, b), c).
Example
Find LCM(12, 18).
- Prime factorization: 12 = 2² × 3 and 18 = 2 × 3². Largest powers: 2² × 3² = 36.
- Using the GCF: GCF(12, 18) = 6, so LCM = 12 × 18 ÷ 6 = 216 ÷ 6 = 36.
Use: to add 1/12 + 5/18, convert both to the denominator 36: 3/36 + 10/36 = 13/36.
Scheduling: one bus every 12 minutes and another every 18 minutes leave together again after 36 minutes.
The LCM calculator finds the LCM of several numbers with steps, the GCF calculator finds the divisor used in the formula, and the factor calculator lists factors for the prime-factorization method.
Common mistakes
- Multiplying the numbers together and stopping. 12 × 18 = 216 is a common multiple, but not the least one.
- Taking the smallest exponent of each prime, which gives the GCF instead.