Richard is making gift bags. He has 36 pencils and 28 pens. How many gift bags can Richard make if there are the same number of pencils and pens in each bag? Use factor trees to solve this problem. How many pencils and pens will be in each bag?

In this concept, you will learn to find the greatest common factor using factor trees.

### Finding the Greatest Common Factor Using Factor Trees

The **greatest common factor (GCF)** is the greatest factor that two or more numbers have in common. The GCF can be found by making a list and comparing all the factors. A factor tree can also be used to find the GCF. The GCF is the product of the common prime factors.

Let’s find the GCF of 20 and 30 using a factor tree.

First, make a factor tree for each number.

Then, identify the common factors. The numbers 20 and 30 have the factors 2 and 5 in common.

Next, multiply the common factors to find the GCF. If there is only one common factor, there is no need to multiply.

The GCF of 20 and 30 is 10.

Note that if the numbers being compared have no factors in common using a factor tree, they still have the factor 1 in common.

### Examples

#### Example 1

Earlier, you were given a problem about Richard who needs to make gift bags with 36 pencils and 28 pens.

Use factor trees to find the most number of bags he can make that have the same number of pencils and pens in each.

First, make a factor tree for each number.

Then, identify the common factors. The common factors are two 2s.

Next, multiply to common factors to find the GCF.

Finally, divide the number of pencils and pens by the GCF, 4.

Richard can make 4 gift bags that have 9 pencils and 7 pens in each bag.

#### Example 2

Find the GCF of 36 and 54 using factor trees.

First, make a factor tree for each number.

Then, identify the common factors. The numbers 36 and 54 have the factors 2 and two 3s in common.

Next, multiply the common factors to find the GCF.

The GCF of 36 and 54 is 18.

#### Example 3

Find the greatest common factor using factor trees.

First, make a factor tree for each number.

Then, identify the common factors. The numbers 14 and 28 have the factors 2 and 7 in common.

Next, multiply the common factors to find the GCF.

The GCF of 14 and 28 is 14.

#### Example 4

Find the greatest common factor using factor trees.

First, make a factor tree for each number.

Then, identify the common factors. The numbers 24 and 34 have the factor 2 in common.

The GCF of 12 and 24 is 12.

#### Example 5

Find the greatest common factor using factor trees.

First, make a factor tree for each number.

Then, identify the common factors. The numbers 19 and 63 have the factor 1 in common.

The GCF of 19 and 63 is 1.

### Review

Find greatest common factor for each pair of numbers.

- 14 and 28
- 14 and 30
- 16 and 36
- 24 and 60
- 72 and 108
- 18 and 81
- 80 and 200
- 99 and 33
- 27 and 117
- 63 and 126
- 89 and 178
- 90 and 300
- 56 and 104
- 63 and 105
- 72 and 128

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.6.