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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With school underway next week or this week for some students, it's important to review material from the previous year. Math is one subject that builds upon previous material. Suffer in something previously and you'll probably struggle in the next class, particularly when it comes to algebra, even sometimes in statistics. We tend to use many of the basics in all math classes. So just be sure to review and if you run into trouble in your math class or have children that run into trouble, get a reliable tutor. Most schools will have a list of tutors. If not, you can find some online on various websites. Or you can consult me, as I've been tutoring for over 14 years now.

Try this little math puzzle.

Suppose you have a four digit number. The first digit is 1/5 of the last digit and the second and third digit together is 3 times the last digit. What is the number? (hint, the sum of the digits is 12)

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The answer it 1155.  The last digit must be a multiple of 5 since the first digit is 1/5. So the last digit must be 5 and the first digit must be 1.  The tricky part at first reading this is that the second and third digit being 3 times the last, you might think the second digit is 15 and the third is also 15, but that makes no sense, because a digit is a single number. the 2nd and third TOGETHER is 15, so 1 in the second digit and 5 in the third. So the answer is 1155.

Here's a simple, but fun activity for kids learning multiplication. Start at the red dot. Answer the multiplication problems in order. The answer will be one of the numbers connected by the black line. Move to that number. If you don't get any of the numbers listed, you are in correct. Keep going through the problems and eventually you will exit at one of the orange blocks. Good luck!

This shows young students how to subtract using borrowing. It's simple mathematics but good for children who are just learning how to subtract.

Here's a fun activity for kids just learning division. Fill in the question marks to complete the division problem.