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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

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.

\begin{align*}20 = \underline{2} \times 2 \times \underline{5} \qquad 30 = \underline{2} \times 3 \times \underline{5}\end{align*}

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

\begin{align*}2 \times 5 = 10\end{align*}

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.

\begin{align*}36 = \underline{2} \times \underline{2} \times 3 \times 3 \qquad 28 = \underline{2} \times \underline{2} \times 7\end{align*}

Next, multiply to common factors to find the GCF.

\begin{align*}2 \times 2 = 4\end{align*}

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

\begin{align*}\begin{array}{rcl} \text{pencils} & = & 36 \div 4 = 9\\ \text{pens} & = & 28 \div 4 = 7 \end{array}\end{align*}

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.

\begin{align*}36 = \underline{2} \times 2 \times \underline{3} \times \underline{3} \qquad 54 = \underline{2} \times \underline{3} \times \underline{3} \times 3\end{align*}

Next, multiply the common factors to find the GCF.

\begin{align*}2 \times 3 \times 3 = 18\end{align*}

The GCF of 36 and 54 is 18.

#### Example 3

Find the greatest common factor using factor trees.

\begin{align*}14\text{ and }28\end{align*}

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.

\begin{align*}14 = \underline{2} \times \underline{7} \qquad 28= \underline{2} \times 2 \times \underline{7}\end{align*}

Next, multiply the common factors to find the GCF.

\begin{align*}2 \times 7 = 14\end{align*}

The GCF of 14 and 28 is 14.

#### Example 4

Find the greatest common factor using factor trees.

\begin{align*}24 \text{ and } 34\end{align*}

First, make a factor tree for each number.

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

\begin{align*}24 = \underline{2} \times 2 \times 2 \times 3 \qquad 34 = \underline{2} \times 17\end{align*}

The GCF of 12 and 24 is 12.

#### Example 5

Find the greatest common factor using factor trees.

\begin{align*}19 \text{ and } 63\end{align*}

First, make a factor tree for each number.

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

\begin{align*}19 = 1 \times 19 \qquad 63 = 3 \times 3 \times 7\end{align*}

The GCF of 19 and 63 is 1.

### Review

Find greatest common factor for each pair of numbers.

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

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

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

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

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

Product

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

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