Calculators > Math > Least Common Multiple (LCM) Calculator

Least Common Multiple (LCM) Calculator

Find the LCM of:
(numbers separated by commas)
For the values: 15, 65, 330
The LCM is: 4290
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What is the Least Common Multiple?

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. 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 = 22 * 31 * 50
  • Prime factorization of 30 = 2 * 3 * 5 = 21 * 31 * 51
  • Using the set of prime numbers from each set with the highest exponent value we take 22 * 31 * 51 = 60
  • Therefore LCM(12,30) = 60.

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

  • Prime factorization of 24 = 2 * 2 * 2 * 3 = 23 * 31 * 50
  • Prime factorization of 300 = 2 * 2 * 3 * 5 * 5 = 22 * 31 * 52
  • Using the set of prime numbers from each set with the highest exponent value we take 23 * 31 * 52 = 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. http://mathworld.wolfram.com/LeastCommonMultiple.html

http://mathforum.org/library/drmath/sets/select/dm_lcm_gcf.html