# LCM Calculator - Least Common Multiple

## Calculator Use

The Least Common Multiple (LCM) is also referred to as the Lowest Common Multiple (LCM) and Least Common Denominator (LCD). For two integers *a* and *b*, denoted LCM(*a,b*), the LCM is the smallest integer that is evenly divisible by both *a* and *b*. For example, LCM(2,3) = 6 and LCM(6,10) = 30. For the least common multiple of more than 2 numbers, say *a, b, c* and *d*, it is the smallest integer that is evenly divisible by all numbers and can be calculated such that LCM(a,b,c,d) = LCM(LCM(LCM(*a,b*),*c*),*d*).

## Least Common Multiple Calculator

Input the numbers you want to evaluate, separated by commas. Do not use comma separatos or decimals within your numbers. For example enter 2500, 1000 rather than 2,500, 1,000.

## How to find the Least Common Multiple

For integers *a* and *b* you can find the greatest common divisor (GCD) of *a* and *b* and use that result to calculate the least common multiple. You can also find the prime factorization of each integer and use those results to calculate the LCM.

### Find LCM by GCD

LCM(a,b) = (a*b)/GCD(a,b)

For example, find the LCM(6,10). First find the GCD(6,10) = 2. Then calculate (6*10)/2 = 60/2 = 30. Therefore, LCM(6,10) = 30.

### Find the LCM by Prime Factorization

The LCM(a,b) is calculated by finding the prime factorization of both a and b then taking the product of the sets of primes with the highest exponent value among a and b.

**For example, for LCM(12,30) we find:**

- Prime factorization of 12 = 2 * 2 * 3 = 2
^{2}* 3^{1}* 5^{0} - Prime factorization of 30 = 2 * 3 * 5 = 2
^{1}* 3^{1}* 5^{1} - Using the set of prime numbers from each set with the highest exponent value we take 2
^{2}* 3^{1}* 5^{1}= 60 - Therefore LCM(12,30) = 60.

**For example, for LCM(24,300) we find:**

- Prime factorization of 24 = 2 * 2 * 2 * 3 = 2
^{3}* 3^{1}* 5^{0} - Prime factorization of 300 = 2 * 2 * 3 * 5 * 5 = 2
^{2}* 3^{1}* 5^{2} - Using the set of prime numbers from each set with the highest exponent value we take 2
^{3}* 3^{1}* 5^{2}= 600 - Therefore LCM(24,300) = 600.

## References

[1] Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st Edition, New York, NY: CRC Press, 2003 p. 101.

[2] Weisstein, Eric W. Least Common Multiple. From *MathWorld*--A Wolfram Web Resource.

The Math Forum: LCM, GCF.

**Cite this content, page or calculator as:**

Furey, Edward "LCM Calculator - Least Common Multiple"; from *http://www.calculatorsoup.com* - Online Calculator Resource.