4-8 Least Common Multiple MAIN IDEA Find the least common multiple of two or more numbers. New Vocabulary multiple least common multiple (LCM) Use cubes to build the first row of each prism as shown. 1. Add a second row to each prism. Record the total number of cubes used in a table like the one shown below. Number of Rows Cubes in Prism A Cubes in Prism B 1 4 6 2 3 8 12 1SJTN" 4 12 18 16 24 Math Online glencoe.com 2. Add rows until each prism has four rows. • Extra Examples • Personal Tutor • Self-Check Quiz 3. Describe two prisms that have the same 1SJTN# number of cubes. See margin. 4. If you keep adding rows, will the two prisms have the same number of cubes again? See margin. A multiple is the product of a number and any whole number. The least common multiple, or LCM, of two or more numbers is the least of their common multiples, excluding zero. Find the LCM 1 Find the LCM of 6 and 10. List the nonzero multiples. METHOD 1 List the multiples of 6 until you come to a number that is also a multiple of 10. multiples of 6: 6, 12, 18, 24, 30, … multiples of 10: 10, 20, 30, … Notice that 30 is also a multiple of 10. The LCM of 6 and 10 is 30. Vocabulary Link Multiply Everyday Use to find the product Multiple Math Use the product of a number and any whole number METHOD 2 6=2·3 10 = 2 · 5 Use prime factorization. The prime factors of 6 and 10 are 2, 3, and 5. The LCM is the least product that contains the prime factors of each number. So, the LCM of 6 and 10 is 2 · 3 · 5 or 30. Lesson 4-8 Least Common Multiple 211_0211_0214_CH04_L8_874046 211 211 9/21/07 9:35:00 AM 2 Find the LCM of 45 and 75. Use Method 2. Find the prime factorization of each number. 45 = 3 · 3 · 5 or 3 2 · 5 75 = 3 · 5 · 5 or 3 · 5 The prime factors of 45 and 75 are 3 and 5. Write the prime factorization using exponents. 2 The LCM is the product of the prime factors 3 and 5, with each one raised to the highest power it occurs in either prime factorization. The LCM of 45 and 75 is 32 · 5 2, which is 225. Find the LCM of each set of numbers. a. 3, 12 b. 10, 12 c. 25, 30 3 PARTY Ling needs to buy paper plates, napkins, and cups for a party. Plates come in packages of 12, napkins come in packages of 16, and cups come in packages of 8. What is the least number of packages she will have to buy if she wants to have the same number of plates, napkins, and cups? First find the LCM of 8, 12, and 16. 8 = 2 · 2 · 2 or 2 3 12 = 2 · 2 · 3 or 2 2 · 3 16 = 2 · 2 · 2 · 2 or 2 4 The prime factors of 8, 12, and 16 are 2 and 3. Write the prime factorization using exponents. The LCM of 8, 12, and 16 is 2 4 · 3, which is 48. To find the number of packages of each Ling needs to buy, divide 48 by the amount in each package. cups: 48 ÷ 8 or 6 packages plates: 48 ÷ 12 or 4 packages napkins: 48 ÷ 16 or 3 packages So, Ling will need to buy 6 packages of cups, 4 packages of plates, and 3 packages of napkins. d. VEHICLES Mr. Hernandez changes his car’s oil every 3 months, rotates the tires every 6 months, and replaces the air filter once a year. If he completed all three tasks in April, what will be the next month he again completes all three tasks? 212 Chapter 4 Fractions, Decimals, and Percents 211_214_C4_L8_892328.indd 212 4/8/10 10:01:06 AM Examples 1–3 (pp. 211–212) Find the LCM of each set of numbers. 28 1. 4, 14 4. 21, 35 Example 3 (p. 212) HOMEWORK For Exercises 8–13, 20 14–19, 21 HELP See Examples 1, 2 3 105 42 3. 12, 15 60 5. 3, 5, 12 60 42 6. 6, 14, 21 7. GOVERNMENT The number of years per term for a U.S. President, senator, and representative is shown. Suppose a senator was elected in the presidential election year 2008. In what year will he or she campaign again during a presidential election year? 2020 Elected Office Term (yr) President 4 Senator 6 Representative 2 Find the LCM for each set of numbers. 8. 6, 8 24 9. 8, 18 72 10. 12, 16 48 315 11. 24, 36 72 12. 11, 12 132 13. 45, 63 14. 2, 3, 5 30 15. 6, 8, 9 72 16. 8, 12, 16 17. 12, 15, 28 Exercise Levels A: 8–21 B: 22–30 C: 31–34 2. 6, 7 420 18. 22, 33, 44 132 48 19. 12, 16, 36 144 20. CHORES Hernando walks his dog every two days. He gives his dog a bath once a week. Today, Hernando walked his dog and then gave her a bath. How many days will pass before he does both chores on the same day? 14 days Friend 21. TEXT MESSAGING Three friends use text messaging to notify their parents of their whereabouts. If all three contact their parents at 3:00 p.m., at what time will all three contact their parents again at the same time? 6:00 P.M. Find the LCM of each set. 22. $3.00, $14.00 $42 23. 10¢, 25¢, 5¢ 50¢ Time Interval Linda every 30 min Brandon every 45 min Edward every 60 min 24. 9 inches, 2 feet 72 in. or 6 ft Write two numbers whose LCM is the given number. 25–28. Sample answers are given. 25. 35 5, 7 26. 56 7, 8 27. 70 10, 35 28. 30 6, 15 29. SNACKS Alvin’s mom needs to buy snacks for soccer practice. Juice boxes EXTRA PRACTICE See pages 678, 707. come in packages of 10. Oatmeal snack bars come in packages of 8. She wants to have the same number of juice boxes and snack bars, what is the least number of packages of each snack that she will have to buy? 4 packages of juice boxes and 5 packages of oatmeal snack bars 30. REASONING The LCM of two consecutive positive numbers is greater than 200 and is a multiple of 7. What are the least possible numbers? 14 and 15 Lesson 4-8 Least Common Multiple 213_0211_0214_CH04_L8_874046 213 213 9/21/07 9:35:10 AM H.O.T. Problems 31. CHALLENGE Two numbers have a GCF of 3 · 5. Their LCM is 22 · 3 · 5. If one 32. Number sense; Sample answer: The LCM of 14, 16, and 8 is 112. So, 112 minutes later, or at 1:27 P.M. 32. SELECT A TECHNIQUE The schedule for of the numbers is 3 · 5, what is the other number? 2 2 · 3 · 5, or 60 Clark Street Train Station each of three trains is shown. Suppose a Train train from each line leaves Clark Street Red-line at 11:35 a.m. Which of the following Blue-line technique(s) might you use to determine Brown-line the next time all three trains will be leaving at the same time? Justify your selection(s). Then use the technique to solve the problem. mental math number sense Leaves Station every 14 minutes every 16 minutes every 8 minutes estimation 33. OPEN ENDED Write three numbers that have an LCM of 30. Sample answer: 3, 10, 15 34. WR ITING IN MATH Describe the relationship between 4, 20, and 5 using the words factor and multiple. 4 and 5 are factors of 20; 20 is a multiple (and LCM) of 4 and 5. 36. SHORT RESPONSE Wil swims every 35. Which rule describes the common multiples of 12 and 18, where n represents the counting numbers? C third day, runs every fourth day, and lifts weights every fifth day. If Wil does all three activities today, how many days will pass before he does all three activities on the same day again? 60 A 12n B 18n C 36n D 216n Write each percent as a decimal. 37. 55% 0.55 38. 26.4% (Lesson 4-7) 0.264 39. _1 % 0.0025 4 0.02 40. 2% 41. DIAMONDS Sixty-eight percent of engagement rings have a diamond that is round in shape. Write this percent as a fraction in simplest form. 42. ALGEBRA Solve 3x = 18. (Lesson 3-3) (Lesson 4-6) 17 _ 25 6 43. ALGEBRA Rose swam 7 laps more than twice the number of laps her sister swam. Write an algebraic expression to represent this situation. (Lesson 3-1) PREREQUISITE SKILL Replace each ● with <, > or = to make a true sentence. 44. 6.85 ● 5.68 214 > 45. 2.34 ● 2.43 < 46. 6.9 ● 5.99 2s + 7 (Page 670) > Chapter 4 Fractions, Decimals, and Percents 214_0211_0214_CH04_L8_874046 214 9/21/07 9:35:14 AM

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