LESSON Page 1 of 5 4.4 Least Common Multiple Now BEFORE WHY? Vocabulary You found the GCF of two numbers. You’ll find the least common multiple of two numbers. So you can design a fitness schedule, as in Ex. 38. multiple, p. 187 common multiple, p. 187 least common multiple (LCM), p. 187 least common denominator (LCD), p. 188 Agriculture Crop rotation is a system in which farmers vary the crops they plant in their fields each year. Suppose a farmer grows alfalfa in a certain field every 6 years. In another field, the farmer grows alfalfa every 10 years. This year, the farmer is growing alfalfa in both fields. In how many years will the farmer grow alfalfa in both fields again? A multiple of a whole number is the product of the number and any nonzero whole number. A multiple that is shared by two or more numbers is a common multiple . Some of the common multiples of 8 and 12 are shown in blue below. Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, . . . Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, . . . The least of the common multiples of two or more numbers is the least common multiple (LCM) . The LCM of 8 and 12 is 24. Example 1 Finding the Least Common Multiple For the crop rotation system described above, the number of years until the farmer grows alfalfa in both fields again is given by the LCM of 6 and 10. You can use one of two methods to find the LCM. Method 1 List the multiples of each number. Identify the least number that is on both lists. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 Multiples of 10: 10, 20, 30, 40, 50, 60 The LCM of 6 and 10 is 30. Method 2 Find the common factors of the numbers. 62 p 3 10 2 p 5 The common factor is 2. Multiply all of the factors, using each common factor only once. LCM 2 p 3 p 5 30 Answer The farmer will grow alfalfa in both fields again in 30 years. Checkpoint Find the least common multiple of the numbers. 1. 16, 24 2. 20, 25 3. 6, 8, 20 Lesson 4.4 4. 15, 30, 50 Least Common Multiple 187 Page 2 of 5 Example 2 Finding the Least Common Multiple of Monomials Find the least common multiple of 9xy 2 and 15x 2y. 9xy 2 3 p 3 p x p y p y 15x 2y 3 p 5 p x p x p y LCM 3 p x p y p 3 p 5 p x p y 45x 2y 2 Study Strategy In Example 2, notice that the LCM of the two monomials includes the higher power of each variable, as well as the higher power of each prime number factor. Common factors are circled and used only once in the LCM. Answer The least common multiple of 9xy 2 and 15x 2y is 45x 2y 2. Least Common Denominator The least common denominator (LCD) of two or more fractions is the least common multiple of the denominators. You can use the LCD to compare and order fractions. Example 3 Comparing Fractions Using the LCD Winter Sports Last year, a winter resort had 144,000 visitors, including 45,000 snowboarders. This year, the resort had 160,000 visitors, including 56,000 snowboarders. In which year was the fraction of snowboarders greater? Solution 1 Write the fractions and simplify. Number of snowboarders Total number of visitors 45,000 144,000 5 16 Number of snowboarders Total number of visitors 56,000 160,000 7 20 Last year: This year: 2 5 16 7 20 Find the LCD of and . The LCM of 16 and 20 is 80. So, the LCD of the fractions is 80. 3 Write equivalent fractions using the LCD. 5 16 5p5 16 p 5 25 80 7 20 Last year: 4 25 80 7p4 20 p 4 28 80 This year: 28 80 5 16 7 20 Compare the numerators: < , so < . Answer The fraction of snowboarders was greater this year. Checkpoint Find the least common multiple of the monomials. 5. 15x 2, 27x 6. 6m2, 10m3 7. 14ab, 21bc 8. r 2, 5rst Use the LCD to determine which fraction is greater. 5 7 9. , 6 9 188 Chapter 4 Factors, Fractions, and Exponents 5 13 10. , 8 20 7 11 11. , 12 15 5 3 12. , 16 10 Page 3 of 5 Example 4 Ordering Fractions and Mixed Numbers 4 33 1 5 10 19 6 Order the numbers 3 , , and from least to greatest. Review Help 1 Write the mixed number as an improper fraction. 4 15 3 p 15 4 15 49 15 3 For help with writing mixed numbers as improper fractions, see p. 778. 2 49 33 15 10 19 6 Find the LCD of , , and . The LCM of 15, 10, and 6 is 30. So, the LCD is 30. 3 Write equivalent fractions using the LCD. 49 49 p 2 98 15 15 p 2 30 4 33 33 p 3 99 10 10 p 3 30 19 19 p 5 95 6 6p5 30 Compare the numerators. 95 98 98 99 19 49 49 33 < and < , so < and < . 30 30 30 30 6 15 15 10 19 6 4 15 33 10 Answer From least to greatest, the numbers are , 3 , and . 4.4 Exercises INTERNET More Practice, p. 806 CLASSZONE.COM eWorkbook Plus Guided Practice Vocabulary Check 1. How are the terms least common multiple and least common denominator related? 4 7 2. Describe how you would use the LCD to compare and . 7 12 Skill Check Find the least common multiple of the numbers. 3. 3, 4 4. 4, 8 5. 18, 24 6. 10, 16 Find the least common multiple of the monomials. 7. 3s, s 2 8. x 4, x 2 9. 15m 2, 9m 10. 8b, 20b 2 Use the LCD to determine which fraction is greater. 3 5 11. , 4 8 2 13 12. , 3 16 3 7 14. , 4 10 2 3 13. , 5 8 15. Error Analysis Describe 16 24 and correct the error in finding the LCM of 16 and 30. 30 2 p 3 p 5 LCM 2 p 3 p 5 480 5 Lesson 4.4 Least Common Multiple 189 Page 4 of 5 Practice and Problem Solving Homework Help Example 1 2 3 4 Exercises 16–27, 36, 38 28–35 39–46 47–54 Online Resources CLASSZONE.COM • More Examples • eTutorial Plus Find the least common multiple of the numbers. 16. 9, 12 17. 3, 8 18. 4, 16 19. 10, 15 20. 21, 14 21. 30, 36 22. 55, 15 23. 42, 66 24. 3, 6, 12 25. 8, 11, 36 26. 10, 12, 14 27. 16, 20, 30 Find the least common multiple of the monomials. 28. 5a 2, 16a 3 29. 21w, 9w 2 30. 17b 2, 3b 3 31. 14x 4, 21x 2 32. 60s4, 24s 3 33. 2n3, 8n2 34. 25a, 40a 2 35. 11s, 33s 2 36. Visual Patterns In the first pattern shown below, the green star repeats every 6 figures. In the second pattern, the green star repeats every 8 figures. How many figures after the first figure will both patterns have a green star? 37. Writing Could you find the greatest common multiple of two numbers? Explain your thinking. 38. Fitness You lift weights every third day and take karate class every Monday. Today you have karate and are lifting weights. In how many days will you next lift weights and have karate on the same day? Use the LCD to determine which fraction is greater. 1 2 39. , 4 7 2 5 40. , 3 8 7 11 41. , 10 15 3 6 42. , 5 11 5 4 43. , 12 15 7 9 44. , 20 25 5 8 45. , 18 21 11 20 46. , 42 63 Order the numbers from least to greatest. 7 11 1 47. , , 1 6 9 3 13 1 27 48. , 3 , 4 2 8 8 1 3 49. , , 15 5 10 5 14 9 50. , , 11 33 22 3 4 7 51. , , 4 9 15 5 7 11 52. , , 6 10 15 5 43 12 53. , 2 , 12 18 5 1 10 13 54. 1, , 1 3 7 33 55. Critical Thinking What is the least number for which the LCM of the number and 12 is 300? Explain your thinking. Find the least common multiple of the monomials. 56. 24de 2, 36d 3e 57. x 3y, 15xy 5 58. 10a 2b 2, 20ab 59. 45gh3, 33g 4h 60. xyz 3, x 2yz 2 61. 26ab2, 28ac3 62. 11rst, 15r 3t 2 63. 30df 2, 40d 3ef 64. Vice Presidents During the period 1800–1900, 6 out of 23 U.S. Vice Presidents later became U.S. Presidents. During the period 1901–2000, 7 out of 21 Vice Presidents later became Presidents. During which period did a greater fraction of Vice Presidents become Presidents? 190 Chapter 4 Factors, Fractions, and Exponents Page 5 of 5 In Exercises 65–68, rewrite the variable expressions with a common denominator. Rewriting Variable Expressions 2a 5b 3 4ab To rewrite and 2 with a common denominator, first find the LCD of the fractions. The LCM of 5b and 4ab 2 is 20ab 2. So, the LCD is 20ab 2. Then write equivalent fractions using the LCD. 2a 2a p 4ab 8a 2b 2 5b 5b p 4ab 20ab 3 3p5 15 2 2 2 4ab 4ab p 5 20ab x y 66. , 6 y 8x x x 65. , 3 4 3x 2 67. 2 , 4y 5xy 3x 5y 68. , 2yz 4xz 69. Critical Thinking Let a and b represent nonzero whole numbers. a 1 a a 2 Find a fraction such that < , < , and b < 30. b 6 b b 7 70. Challenge Copy and complete the table for the given values of a and b. Describe any relationships you notice between the product of the LCM and the GCF and the product of a and b. Prime factorizations LCM GCF LCM p GCF a p b Given numbers Mixed Review a 6, b 18 ? ? ? ? ? a 15, b 35 ? ? ? ? ? a 6, b 20 ? ? ? ? ? a 12, b 60 ? ? ? ? ? Evaluate the expression when n 5. (Lesson 1.2) 71. n2 72. n3 73. n4 74. n5 Write the prime factorization of the number. (Lesson 4.1) 75. 28 76. 39 77. 81 78. 165 79. Cookies You are making gift boxes filled with cookies to give to friends. You have 64 peanut butter cookies, 80 chocolate chip cookies, and 56 sugar cookies. What is the greatest number of identical gift boxes that you can make? (Lesson 4.2) Standardized Test Practice 80. Multiple Choice Which expression is the least common multiple of the monomials 27w 4z and 75w 2z 2? A. 3w 2z B. 75w 4z 2 C. 675w 2z D. 675w 4z 2 81. Multiple Choice Which list shows the fractions in order from least to greatest? 2 1 4 F. 9, 6, 25 3 11 9 G. 7, 24, 21 7 3 5 H. 20, 8, 12 Lesson 4.4 2 19 21 I. 5, 40, 45 Least Common Multiple 191

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