Quick Guide to Finding the Greatest Common Factor

A factor is a number that divides evenly into another number. For example, 2 divides evenly into 12, so 2 is a factor of 12. Finding the factors is known as factoring. Suppose we want to find the largest number that is a factor or two or more numbers. That number is called the greatest common factor (GCF) of the numbers.
To find the greatest common factor, do the prime factorization of each number. Then find the lowest power of common factors and multiply them together. But what is meant by prime factorization and how is this done?
When each factor of a number is a prime number, the number is said to be in prime factored form, or prime factorization.

Examples:
Find the prime factored form (or prime factorization) of the following:
1. 24 Divide 24 by 2 to get 12. Then 12 can be divided by 2 to get 6. Next, 6 can be divided by 2 to get 3. That leaves us with 2 x 2 x 2 x 3. Therefore, the prime factorization is 2³ x 3.
2. 72
Divide 72 by 2 to get 36. Then 36 can be divided by 2 to get 18. Next, 18 can be divided by 2 to get 9. Finally, 9 can be divided by 3 to get 3. That leaves us with 2 x 2 x 2 x 3 x 3. Therefore, the prime factorization is 2³ x 3².
Notice how multiples of the same factor are written in exponential form. Note that if you are unsure what number divides evenly into another number, if the number is even it is divisible by 2. If the sum of the digits in the number is divisible by 3, then the entire number is divisible by 3. (18... 1 + 8 = 9, 9 is divisible by 3, so 18 is divisible by 3.
Now that we understand how to do prime factorization, we can find the greatest common factor.

Example:
Find the greatest common factor of 40 and 72.
40/5 = 8, so we have 5 x 8 as factors of 40. Notice that 8 can be factored into 2 x 2 x 2. Therefore, the prime factorization of 40 is 5 x 2 x 2 x 2. The prime factorization of 72 is 2 x 2 x 2 x 3 x 3 from the second example above. Notice the common factors are 2 x 2 x 2, so the greatest common factor is 8.

Example:
Find the greatest common factor of 45 and 135.
45/5 = 9, so we have 5 x 9 as factors of 45. Notice that 9 is 3 x 3. Therefore, the prime factorization of 45 is 5 x 3 x 3.
135/5 = 27, so we have 5 x 27 as factors of 135. Notice that 27 is 3 x 3 x 3. Therefore, the prime factorization of 135 is 5 x 3 x 3 x 3.
Notice the common factors are 3 and 5. The lowest power of each factor gives us 3 x 3 x 5. So, the greatest common factor is 45.

Example: Find the GCF of 6 a² b , 24 a² b² and 48 a³ b³ . Note that I am using * as the multiplication symbol instead of "x" for more clarity.
6a²b = 3 * 2 * a * a * b
24a²b² = 3 * 2 * 2 * 2 * a * a * b * b
48a³b³ = 3 * 2 * 2 * 2 * 2 * a * a * a * b * b * b
Notice the common factors are 3, 2, a, and b. The lowest power of each factor gives us 3 * 2 * a * a * b. Therefore, the greatest common factor is 6a²b.

Example:
Find the GCF of 15x²y³, 20y², and 45xy.
15x²y³ = 3 * 5 * x * x * y * y * y
20y² = 2 * 2 * 5 * y * y
45xy = 3 * 3 * 5 * x * y
Notice the common factors are 5 and y. The term 20y² has 2 as a factor but the other two don't, and 45xy and 15x²y³ have 3 and x as a factor, but 20y² doesn't. The lowest power of each factor is 5 and y, therefore the greatest common factor is 5y.
This quick guide with examples on prime factorization and finding the greatest common factor should assist any student having difficulty on these topics.
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