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4-8 Least Common Multiple MAIN IDEA Find the least common Use cubes to build the first row of each prism as shown. multiple of two or more 1. Add a second row to each prism. Record numbers. the total number of cubes used in a table New Vocabulary like the one shown below. multiple Number of Rows 1 2 3 4 least common multiple Cubes in Prism A 4 8 12 16 (LCM) Cubes in Prism B 6 12 18 24 Math Online glencoe.com 2. Add rows until each prism has four rows. • Extra Examples 3. Describe two prisms that have the same • Personal Tutor • Self-Check Quiz 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. METHOD 1 List the nonzero multiples. 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. METHOD 2 Use prime factorization. Vocabulary Link Multiply 6=2·3 The prime factors of 6 Everyday Use to find the 10 = 2 · 5 and 10 are 2, 3, and 5. product Multiple Math Use the product of The LCM is the least product that contains the prime factors of each a number and any whole number. So, the LCM of 6 and 10 is 2 · 3 · 5 or 30. number Lesson 4-8 Least Common Multiple 211 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 The prime factors of 45 and 75 2 are 3 and 5. Write the prime 75 = 3 · 5 · 5 or 3 · 5 factorization using exponents. 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 The prime factors of 8, 12, and 4 16 are 2 and 3. Write the prime 16 = 2 · 2 · 2 · 2 or 2 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 Examples 1–3 Find the LCM of each set of numbers. (pp. 211–212) 1. 4, 14 28 2. 6, 7 42 3. 12, 15 60 4. 21, 35 105 5. 3, 5, 12 60 6. 6, 14, 21 42 Example 3 7. GOVERNMENT The number of years per term for a Elected Office Term (yr) (p. 212) U.S. President, senator, and representative is shown. President 4 Suppose a senator was elected in the presidential Senator 6 election year 2008. In what year will he or she Representative 2 campaign again during a presidential election year? 2020 Find the LCM for each set of numbers. HOMEWORK HELP For See 8. 6, 8 24 9. 8, 18 72 10. 12, 16 48 Exercises Examples 11. 24, 36 72 12. 11, 12 132 13. 45, 63 315 8–13, 20 1, 2 14–19, 21 3 14. 2, 3, 5 30 15. 6, 8, 9 72 16. 8, 12, 16 48 17. 12, 15, 28 420 18. 22, 33, 44 132 19. 12, 16, 36 144 Exercise Levels 20. CHORES Hernando walks his dog every A: 8–21 two days. He gives his dog a bath once a B: 22–30 week. Today, Hernando walked his dog C: 31–34 and then gave her a bath. How many days will pass before he does both Friend Time Interval chores on the same day? 14 days Linda every 30 min Brandon every 45 min 21. TEXT MESSAGING Three friends use text Edward every 60 min 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¢ 24. 9 inches, 2 feet 72 in. or 6 ft Write two numbers whose LCM is the given number. 25–28. Sample 25. 35 5, 7 26. 56 7, 8 27. 70 10, 35 28. 30 6, 15 answers are given. 29. SNACKS Alvin’s mom needs to buy snacks for soccer practice. Juice boxes 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 EXTRA PRACTICE 30. REASONING The LCM of two consecutive positive numbers is greater than See pages 678, 707. 200 and is a multiple of 7. What are the least possible numbers? 14 and 15 Lesson 4-8 Least Common Multiple 213 H.O.T. Problems 31. CHALLENGE Two numbers have a GCF of 3 · 5. Their LCM is 22 · 3 · 5. If one of the numbers is 3 · 5, what is the other number? 2 2 · 3 · 5, or 60 32. Number 32. SELECT A TECHNIQUE The schedule for Clark Street Train Station sense; Sample each of three trains is shown. Suppose a Train Leaves Station answer: The LCM train from each line leaves Clark Street Red-line every 14 minutes of 14, 16, and 8 is at 11:35 a.m. Which of the following Blue-line every 16 minutes 112. So, 112 technique(s) might you use to determine minutes later, or the next time all three trains will be leaving Brown-line every 8 minutes at 1:27 P.M. at the same time? Justify your selection(s). Then use the technique to solve the problem. mental math number sense 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. 35. Which rule describes the common 36. SHORT RESPONSE Wil swims every multiples of 12 and 18, where n third day, runs every fourth day, represents the counting numbers? C and lifts weights every fifth day. A 12n If Wil does all three activities today, how many days will pass before he B 18n does all three activities on the same C 36n day again? 60 D 216n Write each percent as a decimal. (Lesson 4-7) 37. 55% 0.55 38. 26.4% 0.264 39. _% 0.0025 1 40. 2% 0.02 4 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. (Lesson 4-6) _ 17 25 42. ALGEBRA Solve 3x = 18. (Lesson 3-3) 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) 2s + 7 PREREQUISITE SKILL Replace each ● with <, > or = to make a true sentence. (Page 670) 44. 6.85 ● 5.68 > 45. 2.34 ● 2.43 < 46. 6.9 ● 5.99 > 214 Chapter 4 Fractions, Decimals, and Percents

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