In algebra, the Least Common Multiple (LCM) of a set of numbers is the smallest number that can be divided evenly by each of the numbers in the set. The Greatest Common Factor (GCM) is the biggest number that is a factor of all the numbers in the set. It is important to remember that the Least Common Multiple is always bigger than the Greatest Common Factor. This is a common cause of confusion.
The first step in finding either the LCM or the GCM of a set of numbers is to factor all the numbers in the set. To factor a number is to ruduce it to the product of prime numbers, or numbers that are only divisible by themselves and 1. For example, 18 can be factored as 3 x 3 x 2.
Factorization can be accomplished by taking a number and dividing it by one of its factors, then further dividing the quotients until it is broken down into indivisible prime numbers. In the illustration here, 525 is factored into 5 x 5 x 7 x 3.
To find the GCF of several numbers, compare their prime factors. Here are the prime factorizations of the numbers 210, 330, and 45.
All three of these numbers share two common prime factors, 3 and 5. Multiplying 3 and 5 together produces the greatest common factor, 15. To find the GCM of two or more numbers, multiply together all the prime factors they have in common.
If a number shares all the prime factors of another number, then the number is a multiple of the second number. The Least Common Multiple is the smallest number (besides 0) that shares all the prime factors of all the numbers in the set. For example, 36 is the LCM of 4 and 9. The prime factorization of 36 is 2 x 2 x 3 x 3. It is the smallest number that shares both the prime factors of 4: 2 x 2, and the prime factors of 9: 3 x 3.
The prime factorizations of 210, 330, and 45 are shown in the picture above. The LCM of these three numbers is 2 x 3 x 3 x 5 x 7 x 11 = 3,080. 3,080 shares all of the prime factors of each of the three numbers. Even though 210 and 330 both have a 2 in their prime factorization, only one copy of 2 is needed in their LCM. However, since there are two copies of 3 in the prime factorization of 45, there must be at least two copies of 3 in the LCM.