# 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 2 integers *a* and *b*, denoted LCM(*a,b*), it 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

Enter the numbers you want evaluated separated by commas, but do not use a thousands separator. Two thousand five hundred and one thousand should be entered as 2500, 1000 not 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 LCM. 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.