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# Greatest Common Factor Using Factor Trees

## Multiplying only the same prime factors on each tree

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Greatest Common Factor Using Factor Trees

Remember Maria from the Find the Greatest Common Factor of Numbers Using Lists? Well, in that Concept, she figured out the basketball teams by using lists. What if she had done it a different way? Could the same result be expected? Let's look at the problem again, but this time we will solve it another way.

The sixth grade teachers have decided to have a big basketball tournament as part of the sixth grade social. The sixth graders in clusters 6A and 6B love basketball, and when the agenda is announced, all of the students are very excited. The biggest question is how many teams to divide the students from each cluster into. The teachers want to have the same number of teams, otherwise it will be difficult to have even games for a tournament.

Cluster 6A has 48 students in it.

Cluster 6B has 44 students in it.

The teachers pose the dilemma to the students and Maria volunteers to figure out the teams. She needs to figure out how many teams to divide each cluster into and how many students will then be on each team. Maria has an idea how to do it. She knows that factors are going to be important. She just isn’t sure how to make certain that each cluster is divided into the same number of teams.

Use this Concept to help Maria solve this problem in a new way.

### Guidance

You just learned how to find the GCF by making lists. We can also find the GCF by making a factor tree.

Find the GCF of 20 and 30.

First, we make a factor tree for each number.

$& \ \ \ 20 && \quad \ \ 30\\& \ \ \big / \ \big\backslash && \quad \ \big / \ \ \big\backslash\\& \ 4 \quad 5 && \quad 5 \ \quad 6\\& \big / \ \big\backslash && \qquad \ \big / \ \big\backslash\\& 2 \ \ 2 && \qquad 3 \ \ \ 2\\& 2^2 \times 5 && 5 \times 3 \times 2$

Here is a tricky one because there is more than one common prime factor. We have both five and two as common factors. When you have more than one common factor, we multiply the common factors to find the GCF.

2 $\times$ 5 $=$ 10

10 is the greatest common factor (GCF).

Now it is your turn. Use factor trees to find the GCF of each pair of numbers.

14 and 28

Solution: 7

12 and 24

Solution: 12

#### Example C

16 and 18

Solution: 2

Now let's go back to Maria and the basketball game. Here is the original problem once again.

The sixth grade teachers have decided to have a big basketball tournament as part of the sixth grade social. The sixth graders in clusters 6A and 6B love basketball, and when the agenda is announced, all of the students are very excited. The biggest question is how many teams to divide the students from each cluster into. The teachers want to have the same number of teams, otherwise it will be difficult to have even games for a tournament.

Cluster 6A has 48 students in it.

Cluster 6B has 44 students in it.

The teachers pose the dilemma to the students and Maria volunteers to figure out the teams. She needs to figure out how many teams to divide each cluster into and how many students will then be on each team. Maria has an idea how to do it. She knows that factors are going to be important. She just isn’t sure how to make certain that each cluster is divided into the same number of teams.

The first thing that we are going to do is to build two factor trees so that we can identify the greatest common factor. The two classes have 48 and 44 students in them. So we can create a factor tree for 44 and one for 48.

$& \ \ \ 48 && \quad \ \ 44\\& \ \ \big / \ \big\backslash && \quad \ \big / \ \ \big\backslash\\& \ 4 \quad 12 && \quad 11 \ \quad 4\\$

The greatest common factor of 48 and 44 is 4.

The students should be divided into 4 teams.

### Vocabulary

Factor
a number multiplied by another number to get a product.
Greatest Common Factor
the greatest factor that two or more numbers has in common.
Product
the answer of a multiplication problem

### Guided Practice

Here is one for you to try on your own.

Find the GCF of 36 and 54 using factor trees.

$& \ \ \ 36 && \quad \ \ 54\\& \ \ \big / \ \big\backslash && \quad \ \big / \ \ \big\backslash\\& \ 4 \quad 9 && \quad 9 \ \quad 6\\$

We could keep on going and factor these values to their primes; however, the greatest common factor has already revealed itself in the first stage of the factor tree.

The greatest common factor of 36 and 54 is 9.

### Practice

Directions: Name the greatest common factor for each pair of numbers. If there isn't a gcf for the pair, then write none.

1. 14 and 28

2. 14 and 30

3. 16 and 36

4. 24 and 60

5. 72 and 108

6. 18 and 81

7. 80 and 200

8. 99 and 33

9. 27 and 117

10. 63 and 126

11. 89 and 178

12. 90 and 300

13. 56 and 104

14. 63 and 105

15. 72 and 128

### Vocabulary Language: English

factor

factor

Factors are the numbers being multiplied to equal a product. To factor means to rewrite a mathematical expression as a product of factors.
Greatest Common Factor

Greatest Common Factor

The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.
Product

Product

The product is the result after two amounts have been multiplied.