Least Common Multiple (LCM)

The least common multiple, or LCM, of two numbers is the smallest number that is divisible by both numbers. To find the LCM, take the prime factorization of both numbers. Then make a list of the "minimum" factors required to obtain both numbers. If the prime factorization of one number contains two 3's and the prime factorization of the other number contains five 3's, write down five 3's.

For example, the least common multiple of 1,575 and 23,100 is 2×2×3×3×5×5×7×11 = 69, 300. 69,300 is divisible by both 1,575 and 23,100, and there is no number smaller than 69,300 that is divisible by both.

Another way to find the LCM is to multiply the two numbers and divide by the GCF. For example, 1, 575×23, 100 = 36, 382, 500. 36, 382, 500/525 = 69, 300. This method is useful when one has a calculator and has already calculated the GCF.

If two numbers are relatively prime, their LCM is the same as their product. Using the second method for calculating the LCM, it is easy to see why this is true. The greatest common factor of two relatively prime numbers is 1, so when the two numbers are multiplied and the result is divided by 1 (the GCF), the result does not change.

The least common multiple of 21 and 40, since they are relatively prime, is 21×40 = 840.

Finding the GCF and LCM for Several Numbers

PARGRAPH It is possible to take the GCF or LCM of more than two numbers. To take the GCF, simply multiply the factors that all the numbers have in common. To take the LCM, multiply the minimum factors required to obtain all the numbers (here, you cannot simply multiply all the numbers and divide by the GCF).