Greatest Common Divisor, and Least Common Multiple

Greatest Common Divisor is also known as the Greatest Common Factor (GCF). You may be given two numbers... let's say 18 and 24, and you are asked to find the GCF of both the numbers. "GCF (18,24) =" Our goal is to find the one factor that both of these numbers have that is the largest. This means that you want to find the one number that divides both of these into a number that has no remainder. There are a few numbers that divide them both, but we are interested in finding the number that is the greatest.

The first thing you need to do is to find the all the possible factors of each number:

18= {1,2,3,6,9,18}

24={1,2,3,4,6,8,12,24}

The factors they have in common are {1,2,3,6} because these are all of the numbers that divide them both without leaving any remainder, and we are looking for the number that is the greatest of them all, and we come up with the number 6 since 6x3=18 and 6x4=24. Therefore, GCF (18,24) = 6

Least Common Multiple

My favorite way to find the Least Common Multiple is a take off of the Factor Tree tool. Let's take a few numbers and find the LCM of them. LCM (18,24,12) One way of doing this could be taking each number, multiplying them by 1, then 2, then 3, etc. etc. until you find them all matching up and finding the LCM. However that is a bit time consuming, so using the Factor Tree to find the prime number factorization of the numbers, we'll find a quick and easy way to find the LCM of 18,24,12.

If we worked the factor tree we would find that the prime factorization would be as follows:

18=32x2

24=3x23 

12=3x22

To easily find the LCM of this set of numbers, we would first take the number 3 because all or one of the numbers has that number in the prime factorization. We would take that number 3 and square it, because that is the largest power of the number 3 that we see. Then we'll take the number 2 since all or one of the numbers contains the number 2. Since the largest one is 2 cubed we will use that. There are no other prime numbers used and we are going to take those numbers and multiply them to find the LCM:

32x23=72

(3x3x2x2x2=72)

This website explains factorization and finding LCM and GCD very well: purple math