Learn to find the greatest common factor of two or more whole numbers.

Course 2

2-7 Greatest Common Factor

Vocabulary

greatest common factor (GCF)

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Course 2

2-7 Greatest Common Factor

Course 2

2-7 Greatest Common Factor

The greatest common factor (GCF) of two or more whole numbers is the greatest whole number that divides evenly into each number.

One way to find the GCF of two or more numbers is to list all the factors of each number. The GCF is the greatest factor that appears in all the lists.

Find the greatest common factor (GCF) of 12, 36, 54.

Additional Example 1: Using a List to Find the GCF

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2-7 Greatest Common Factor

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54

The GCF is 6.

List all of the factors of each number.

Circle the greatest factor that is in all the lists.

Check It Out: Example 1

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2-7 Greatest Common Factor

Find the greatest common factor of 14, 28, 63.

14: 1, 2, 7, 14

28: 1, 2, 4, 7, 14, 28

63: 1, 3, 7, 9, 21, 63

The GCF is 7.

List all of the factors of each number.

Circle the greatest factor that is in all the lists.

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2-7 Greatest Common Factor

A Different Way to Find GCF

Another way to find the GCF of two or more numbers is to use the prime factorization of the numbers. Find the common prime factors. If there is more than one common prime factor you will need to multiply them to get the GCF.

36 = 2 · 2 · 3 · 3

42 = 2 · 3 · 72 · 3 = 6

The GCF is 6

Find the greatest common factor (GCF).

Additional Example 2A: Using Prime Factorization to Find the GCF

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2-7 Greatest Common Factor

40, 56

40 = 2 · 2 · 2 · 5

56 = 2 · 2 · 2 · 7

2 · 2 · 2 = 8

The GFC is 8.

Write the prime factorization of each number and circle the common prime factors.

Multiply the common prime factors.

Find the greatest common factor (GCF).

Additional Example 2B: Using Prime Factorization to Find the GCF

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2-7 Greatest Common Factor

252, 180, 96, 60

252 = 2 · 2 · 3 · 3 · 7

180 = 2 · 2 · 3 · 3 · 5

96 = 2 · 2 · 2 · 2 · 2 · 3

60 = 2 · 2 · 3 · 5

2 · 2 · 3 = 12

The GCF is 12.

Write the prime factorizationof each number and circlethe common prime factors.

Multiply the common primefactors.

Check It Out: Example 2A

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2-7 Greatest Common Factor

Find the greatest common factor (GCF).

72, 84

72 = 2 · 2 · 2 · 3 · 3

84 = 2 · 2 · 7 · 3

2 · 2 · 3 = 12

The GCF is 12.

Write the prime factorization of each number and circle the common prime factors.Multiply the common prime factors.

Check It Out: Example 2B

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2-7 Greatest Common Factor

Find the greatest common factor (GCF).

360, 250, 170, 40

360 = 2 · 2 · 2 · 3 · 3 · 5

250 = 2 · 5 · 5 · 5

170 = 2 · 5 · 17

40 = 2 · 2 · 2 · 5

2 · 5 = 10

The GCF is 10.

Write the prime factorizationof each number and circle the common prime factors.

Multiply the common primefactors.

You have 120 red beads, 100 white beads, and 45 blue beads. You want to use all the beads to make bracelets that have red, white, and blue beads on each. What is the greatest number of matching bracelets you can make?

Additional Example 3: Problem Solving Application

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2-7 Greatest Common Factor

Additional Example 3 Continued

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2-7 Greatest Common Factor

11 Understand the Problem

Rewrite the question as a statement.

• Find the greatest number of matching bracelets you can make.

List the important information:

• There are 120 red beads, 100 white beads, and 45 blue beads.

• Each bracelet must have the same number of red, white, and blue beads.

The answer will be the GCF of 120, 100, and 45.

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2-7 Greatest Common Factor

22 Make a Plan

You can list the prime factors of 120, 100,and 45 to find the GFC.

Solve33

120 = 2 · 2 · 2 · 3 · 5

100 = 2 · 2 · 5 · 5

45 = 3 · 3 · 5

The GFC of 120, 100, and 45 is 5.

You can make 5 bracelets.

Additional Example 3 Continued

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2-7 Greatest Common Factor

Look Back44

If you make 5 bracelets, each one will have 24 red beads, 20 white beads, and 9 bluebeads, with nothing left over.

Additional Example 3 Continued

Check It Out: Example 3

Nathan has made fishing flies that he plans to give away as gift sets. He has 24 wet flies and 18 dry flies. Using all of the flies, how many sets can he make?

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2-7 Greatest Common Factor

Check It Out: Example 3 Continued

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2-7 Greatest Common Factor

11 Understand the Problem

Rewrite the question as a statement.

• Find the greatest number of sets of flies he can make.

List the important information:

• There are 24 wet flies and 18 dry flies. • He must use all of the flies.

The answer will be the GCF of 24 and 18.

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2-7 Greatest Common Factor

22 Make a Plan

You can list the prime factors of 24 and 18 to find the GCF.

Check It Out: Example 3 Continued

Solve33

24 = 2 · 2 · 2 · 3

18 = 2 · 3 · 3

You can make 6 sets of flies.

2 · 3 = 6Multiply the prime factors that are common to both 24 and 18.

Check It Out: Example 3 Continued

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2-7 Greatest Common Factor

Look Back44

If you make 6 sets, each set will have 3 dry flies and 4 wet flies.

Lesson Quiz: Part I

Find the greatest common factor (GCF).

1. 28, 40

2. 24, 56

3. 54, 99

4. 20, 35, 70

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Lesson Quiz: Part II

5. The math clubs from 3 schools agreed to a competition. Members from each club must be divided into teams, and teams from all clubs must be equally sized. What is the greatest number of members that can be on a team if Georgia has 16 members, William has 24 members, and Fulton has 72 members?

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