Page 1 of 4 Least Common Multiple BEFORE Now WHY? You found greatest common factors. You’ll find least common multiples. So you can coordinate schedules, as with the soccer games in Ex. 7. In the Real World Word Watch multiple, p. 235 common multiple, p. 235 least common multiple (LCM), p. 236 Ferry Boats Two ferry boats leave a loading platform at the same time. One of the ferry boats returns to the loading platform every 25 minutes. The other returns every 30 minutes. In the next 300 minutes, when will they return at the same time? You can use multiples to answer the question above. A multiple of a number is the product of the number and any nonzero whole number. The three dots show that the pattern continues forever. Multiples of 2: 2, 4, 6, 8, 10, 12, 14, . . . A multiple shared by two or more numbers is a common multiple . EXAMPLE 1 Finding a Common Multiple You can use common multiples to answer the question above about ferry boats. Begin by writing the multiples of 25 and 30. Then identify common multiples through 300. Multiples of 25: 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300 Multiples of 30: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300 The common multiples of 25 and 30 are 150 and 300. ANSWER The ferry boats will return to the loading platform at the same time in 150 minutes and in 300 minutes. Your turn now 1. 2, 3 Find two common multiples of the numbers. 2. 3, 5 3. 8, 10 4. 6, 18 5. A cuckoo clock has birds that pop out of their nests every 6 minutes and dancers that pop out every 15 minutes. Suppose that the birds and dancers have just popped out at the same time. When will this happen again in the next 60 minutes? Lesson 5.4 Least Common Multiple 235 Page 2 of 4 Finding the Least Common Multiple (LCM) The least common multiple of two or more numbers is the smallest of the common multiples. Below are two methods to find the LCM. Method 1: Start listing the multiples of each number. Then find the smallest of the common multiples. Method 2: Write the prime factorizations of the numbers. Multiply together the prime factors, using each prime factor the greatest number of times it is a factor of any of the numbers. EXAMPLE with 2 Finding the LCM Find the LCM of 9 and 12. Solving If the only common factor of two numbers is 1, then their least common multiple is the product of the two numbers. Multiples of 9: 9, 18, 27, 36, 45, 54, . . . Multiples of 12: 12, 24, 36, 48, . . . ANSWER The LCM of 9 and 12 is 36. EXAMPLE 3 Using Prime Factorization Find the LCM of 42 and 60 using prime factorization. 1 Write the prime factorizations. Circle any common factors. 42 2 3 7 60 2 2 3 5 2 Multiply together the prime factors, using each circled factor the greatest number of times it occurs in either factorization. 2 2 3 5 7 420 ANSWER The LCM of 42 and 60 is 420. 236 Chapter 5 Your turn now Find the LCM of the numbers. 6. 8, 12 7. 7, 8, 14 Number Patterns and Fractions 8. 50, 90 Page 3 of 4 INTERNET Exercises eWorkbook Plus CLASSZONE.COM More Practice, p. 712 Getting Ready to Practice Vocabulary List the first three multiples of the number. 1. 7 2. 8 3. 11 4. 16 Use the prime factorizations to find the LCM of the numbers. 5. 32 2 2 2 2 2 6. 48 2 2 2 2 3 24 2 2 2 3 56 2 2 2 7 7. Guided Problem Solving You bring the drinks for your soccer team every sixth game. Every third game is a home game. When will you first bring the drinks to a home game? If there are 20 games in a season, how many times will you bring the drinks to a home game this season? 1 List the multiples of 6 and 3. 2 Find the common multiples of 6 and 3 from 1 to 20. 3 Use the results of Step 2 to answer the questions being asked. Practice and Problem Solving Find the LCM of the numbers by listing multiples. 8. 3, 7 12. 3, 6, 9 9. 5, 8 13. 4, 8, 16 16. Find the Error Describe and correct the error in finding the LCM of 24 and 36. with Example 1 2 3 10. 6, 10 11. 10, 12 14. 2, 3, 5 15. 4, 9, 24 24 2 2 2 3 36 2 2 3 3 The LCM of 24 and 36 is 2 2 3 12. Homework Exercises 8–15, 25 8–15, 25 16–25 Online Resources CLASSZONE.COM • More Examples • eTutorial Plus Find the LCM of the numbers using prime factorization. 17. 21, 28 18. 30, 42 19. 22, 36 20. 32, 40 21. 27, 45 22. 56, 64 23. 60, 72 24. 50, 75 25. Writing A store gives every 20th customer a $20 gift certificate. Every 75th customer gets a $75 gift certificate. Which customer will be the first to receive both types of gift certificates? Explain how you found your answer. Lesson 5.4 Least Common Multiple 237 Page 4 of 4 26. Insects A cricket and a grasshopper are in a jumping contest. The cricket jumps three inches every jump and the grasshopper jumps four inches every jump. Name 4 points where the cricket and the grasshopper will both land. How many jumps will it take each of them to land at these points? Number Sense Find a pair of numbers that matches the description. 27. The LCM of two prime numbers is 51. 28. The LCM of two numbers is 48. Their sum is 19. 29. The LCM of two numbers is 16. Their product is 64. 30. Challenge Find the GCF and the LCM of 6 and 12. How does the product of the GCF and the LCM compare to the product of 6 and 12? Try several pairs of numbers. What does this suggest about the product of two whole numbers and the product of their GCF and LCM? Mixed Review 31. Write two and fifty-six thousandths as a decimal. (Lesson 3.1) Find the quotient. (Lesson 4.6) 32. 20.44 0.56 33. 13.08 0.24 34. 1.76 5.5 35. 1.8 7.2 Write two fractions that are equivalent to the given fraction. (Lesson 5.3) 1 36. 8 3 37. 10 5 38. 12 12 39. 17 Basic Skills Copy and complete the statement with <, >, or . 40. 416 _?_ 419 41. 680 _?_ 68 42. 32 _?_ 352 43. 36 _?_ 390 Test-Taking Practice INTERNET State Test Practice CLASSZONE.COM 44. Multiple Choice What is the least common multiple of 10 and 3? A. 1 B. 13 C. 30 D. 60 45. Short Response Pencils come in packages of 10. Rulers come in packages of 8. Hannah wants exactly one pencil for every ruler. What is the smallest number of packages of each she will need to buy? Explain how you found your answer. 238 Chapter 5 Number Patterns and Fractions

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