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5-2 Greatest Common Factor Name Date Here are two ways to find the GCF of 54 and 90: • List all the factors of each number, then choose the greatest common factor. Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54 Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 Common factors: 1, 2, 3, 6, 9, 18. The GCF is 18. • Find the prime factorization of each number, then multiply the common factors. 54 = 2 • 3 • 3 • 3 90 = 2 • 3 • 3 • 5 Multiply the common Remember: Every number factors: 2 • 3 • 3 = 18. has 1 as a factor. A prime number has exactly So the GCF is 18. two factors, itself and 1. A fraction is in simplest form when its numerator and denominator have a GCF of 1. Here are two ways to write 54 90 in simplest form: Divide the numerator and denominator by the GCF. 54 18 90 18 Use prime factorization. 54 90 35 1 1 1 1 1 1 22 •• 33 •• 33 •• 53 35 To write equivalent fractions for 54 90 , multiply or divide both numerator and denominator by the same nonzero number. •2 108 Multiply. 54 90 • 2 180 6 9 Divide. 54 10 90 9 List the factors of each number. Then find the GCF for each pair of numbers. 1, 2, 3, 4, 6, 8, 12, 24 1. 24 1, 2, 3, 6, 9, 18 18 20 6 GCF: GCF: 1, 2, 3, 6, 7, 14, 21, 42 3. 42 36 14 GCF: 1, 2, 3, 4, 6, 9, 12, 18, 36 1, 2, 4, 5, 10, 20 4 4. 96 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 1, 2, 7, 14 14 Copyright © by William H. Sadlier, Inc. All rights reserved. 2. 36 GCF: 1, 2, 3, 4, 6, 9, 12, 18, 36 12 Find the GCF. Use prime factorization. 5. 30 and 54 6. 27 and 90 7. 14 and 28 30 = 2 • 3 • 5 54 = 2 • 3 • 3 • 3 27 = 3 • 3 • 3 90 = 2 • 3 • 3 • 5 14 = 2 • 7 28 = 2 • 2 • 7 GCF: 2 • 3 = 6 GCF: 9 GCF: 14 8. 35 and 28 9. 17 and 19 10. 43 and 13 35 = 5 • 7 28 = 2 • 2 • 7 17 = 1 • 17 19 = 1 • 19 43 = 1 • 43 13 = 1 • 13 GCF: 7 GCF: 1 GCF: 1 11. 25 and 50 12. 50 and 75 13. 8 and 20 25 = 5 • 5 50 = 2 • 5 • 5 50 = 2 • 5 • 5 75 = 3 • 5 • 5 8=2•2•2 20 = 2 • 2 • 5 GCF: 25 GCF: 25 GCF: 4 Lesson 5-2, pages 110–111. Chapter 5 125 For More Practice Go To: Find the GCF. List the factors or use prime factorization. 14. 9, 24, 27 15. 15, 45, 60 GCF: 3 16. 13, 15, 22 GCF: 15 18. 36, 63, 72 GCF: 1 19. 64, 48, 32 GCF: 9 17. 56, 72, 40 GCF: 8 20. 48, 60, 84 GCF: 16 21. 42, 75, 90 GCF: 12 GCF: 3 Write each fraction in simplest form. Use the GCF or prime factorization. 6 22. 18 23. 12 24 646 1846 24. 15 20 12 4 12 24 4 12 5 13 26. 36 45 15 4 5 20 4 5 1 52 27. 35 90 2•2•3•3 3•3•5 4 16 4 4 20 4 4 3 54 28. 42 56 5•7 2•3•3•5 55 25. 16 20 7 5 18 4 55 39 29. 104 2•3•7 2•2•2•7 3 • 13 2 • 2 • 2 • 13 3 54 3 58 Write two equivalent fractions for each given fraction. Possible answers shown. 30. 12 32 31. 25 70 12 32 3 32 32. 112 25 70 24 5 8 5 64 34. 102 316 5 50 102 316 51 204 5 158 5 632 32 112 5 14 5 140 35. 14 21 33. 108 144 16 2 36. 24 90 14 21 2 5 72 5 4 54 3 55 105 5 21 5 210 11 110 55 37. 105 24 90 28 5 3 5 42 108 144 5 56 5 7 12 48 5 45 5 180 38. Jesse says that all these fractions are in simplest form. Nita says that one is not. Who is correct? 8 2 5 11 9 39. Sergio says that he can quickly find the GCF of any two consecutive numbers without knowing any of the factors of either number. What is his secret? 9 11 17 12 39 3 9 Nita is right, since 39 5 13. The GCF of two consecutive numbers is 1. 40. Sometimes a pair of composite numbers has 1 as their only common factor. Find a value of n that is a composite number less than 72 that makes this statement true. Explain how you found the number. For example, if the fraction n is in simplest form, then the GCF of n and 72 is 1. 72 25 Possible response: If n 5 25, then 72 is in simplest form. The prime factorization of the composite number 25 is 52. The prime factorization of 72 is 23 • 32. The only common factor of the numerator and denominator is 1, so the fraction is in simplest form. 126 Chapter 5 Copyright © by William H. Sadlier, Inc. All rights reserved. Solve. Check to justify your answer. Check students’ work.