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EXPLORATION 2
EXPLORATION 1
EXPLORATION 2
EXPLORATION 1
Section 1.4 Greatest Common Factor 21
Greatest Common FactorGreatest Common Factor
Using Prime Factors
Work with a partner. a. Each Venn diagram represents the prime factorizations of two numbers. Identify each pair of numbers. Explain your reasoning.
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3 3 3
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2
3 3
5 11
b. Create a Venn diagram that represents the prime factorizations of 36 and 48. c. Repeat part (b) for the remaining number pairs in Exploration 1. d. STRUCTURE Make a conjecture about the relationship between the greatest common factors you found in Exploration 1 and the numbers in the overlaps of the Venn diagrams you just created.
Identifying Common Factors
Work with a partner. In parts (a)–(d), create a Venn diagram that represents the factors of each number and identify any common factors. a. 36 and 48 b. 16 and 56 c. 30 and 75 d. 54 and 90 e. Look at the Venn diagrams in parts (a)–(d). Explain how to identify the greatest common factor of each pair of numbers. Then circle it in each diagram.
Learning Target: Find the greatest common factor of two numbers. Success Criteria: • I can explain the meaning of factors of a number.
- I can use lists of factors to identify the greatest common factor of numbers.
- I can use prime factors to identify the greatest common factor of numbers.
A Venn diagram uses circles to describe relationships between two or more sets. The Venn diagram shows the factors of 12 and 15. Numbers that are factors of both 12 and 15 are represented by the overlap of the two circles.
Interpret a Solution What does the diagram representing the prime factorizations mean?
Math Practice
Factors of 12
Factors of 15 2 4 6
1 3 15
5 12
22 Chapter 1 Numerical Expressions and Factors
1.4 Lesson
Multi-Language Glossary atBigIdeasMath.com
Key Vocabulary Venn diagram, p. 21 common factors, p. 22 greatest common factor, p. 22
Factors that are shared by two or more numbers are called common factors. Th e greatest of the common factors is called the greatest common factor (GCF). One way to find the GCF of two or more numbers is by listing factors.
Find the GCF of 24 and 40. List the factors of each number. Factors of 24: 1 , 2 , 3, 4 , 6, 8 , 12, 24 Circle the common factors. Factors of 40: 1 , 2 , 4 , 5, 8 , 10, 20, 40 Th e common factors of 24 and 40 are 1, 2, 4, and 8. The greatest of these common factors is 8.
So, the GCF of 24 and 40 is 8.
Try It Find the GCF of the numbers using lists of factors.
1. 8, 36 2. 18, 72 3. 14, 28, 49
EXAMPLE 1 Finding the GCF Using Lists of Factors
Another way to find the GCF of two or more numbers is by using prime factors. Th e GCF is the product of the common prime factors of the numbers.
Find the GCF of 12 and 56. Make a factor tree for each number. 12 (^2) ⋅ 6 (^2) ⋅ 3
(^7) ⋅ 8 (^2) ⋅ 4 (^2) ⋅ 2 Write the prime factorization of each number.
12 = (^2) ⋅ 2 ⋅ 3 Circle the common prime factors. 56 = (^2) ⋅ 2 ⋅ 2 ⋅ 7
(^2) ⋅ 2 = 4 Find the product of the common prime factors.
So, the GCF of 12 and 56 is 4.
Try It Find the GCF of the numbers using prime factorizations.
4. 20, 45 5. 32, 90 6. 45, 75, 120
EXAMPLE 2^ Finding the GCF Using Prime Factorizations
Examples 1 and 2 show two different methods for fi nding the GCF. After solving with one method, you can use the other method to check your answer.
24 Chapter 1 Numerical Expressions and Factors
Self-Assessment for Problem Solving Solve each exercise. Then rate your understanding of the success criteria in your journal.
13. You use 30 sandwiches and 42 granola bars to make identical picnic baskets. You make the greatest number of picnic baskets with no food left over. How many sandwiches and how many granola bars are in each basket? 14. You fill bags with cookies to give to your friends. You bake 45 chocolate chip cookies, 30 peanut butter cookies, and 15 oatmeal cookies. You want identical groups of cookies in each bag with no cookies left over. What is the greatest number of bags you can make?
You are filling piñatas for your friend’s birthday party. The list shows the gifts you are putting into the piñatas. You want identical groups of gifts in each piñata with no gifts left over. What is the greatest number of piñatas you can make? Th e GCF of the numbers of gifts represents the greatest number of identical groups of gifts you can make with no gifts left over. So, to find the number of piñatas, find the GCF. Write the prime factorization of each number.
18 = (^2) ⋅ 3 ⋅ 3 24 = (^2) ⋅ 3 ⋅ 2 ⋅ 2 Circle the common prime factors. 42 = (^2) ⋅ 3 ⋅ 7
(^2) ⋅ 3 = 6 Find the product of the common prime factors.
Th e GCF of 18, 24, and 42 is 6.
So, you can make at most 6 piñatas.
EXAMPLE 4 Modeling Real Life
Th
Check Verify that 6 identical piñatas will use all of the gifts. 18 kazoos ÷ 6 piñatas = 3 kazoos per piñata 24 mints ÷ 6 piñatas = 4 mints per piñata 42 lollipops ÷ 6 piñatas = 7 lollipops per piñata ✓
Section 1.4 Greatest Common Factor 25
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1.4 Practice
Review & Refresh
List the factor pairs of the number.
1. 20 2. 16 3. 56 4. 87
Tell whether the statement is always, sometimes, or never true.
5. A rectangle is a rhombus. 6. A rhombus is a square. 7. A square is a rectangle. 8. A trapezoid is a parallelogram.
Concepts, Skills, & Problem Solving
USING A VENN DIAGRAM Use a Venn diagram to find the greatest common factor of the numbers. (See Exploration 1, p. 21.)
9. 12, 30 10. 32, 54 11. 24, 108
FINDING THE GCF Find the GCF of the numbers using lists of factors.
12. 6, 15 13. 14, 84 14. 45, 76 15. 39, 65 16. 51, 85 17. 40, 63 18. 12, 48 19. 24, 52 20. 30, 58
FINDING THE GCF Find the GCF of the numbers using prime factorizations.
21. 45, 60 22. 27, 63 23. 36, 81 24. 72, 84 25. 61, 73 26. 38, 95 27. 60, 75 28. 42, 60 29. 42, 63 30. 24, 96 31. 189, 200 32. 90, 108
OPEN-ENDED Write a pair of numbers with the indicated GCF.
33. 5 34. 12 35. 37 36. MODELING REAL LIFE A teacher is making identical activity packets using 92 crayons and 23 sheets of paper. What is the greatest number of packets the teacher can make with no items left over? 37. MODELING REAL LIFE You are making balloon arrangements for a birthday party. There are 16 white balloons and 24 red balloons. Each arrangement must be identical. What is the greatest number of arrangements you can make using every balloon?
