Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

Warm Up Write the prime factorization of each number. 1. 68 2. 225 3. 940 Find the greatest common factor. 4. 27 and 45 5. 32 and 80 22 · 17 32 · 52 22 · 5 · 47 9 16

Problem of the Day Franklin had some counters. When he counted them by twos, threes, fours, and fives, he had one left over. If Franklin had more than 100 and fewer than 150 counters how many did he have? 121

Learn to find the least common multiple of two or more whole numbers.

Vocabulary multiple least common multiple (LCM)

The maintenance schedule on Ken’s pickup truck shows that the tires should be rotated every 7,500 miles and that the oil filter should be replaced every 5,000 miles. What is the lowest mileage at which both services are due at the same time? To find the answer you can use least common multiples.

A multiple of a number is a product of that number and a whole number A multiple of a number is a product of that number and a whole number. Some multiples of 7,500 and 5,000 are as follows: 7,500: 7,500, 15,000, 22,500, 30,000, 37,500, 45,000, . . . 5,000: 5,000, 10,000, 15,000, 20,000, 25,000, 30,000, . . . A common multiple of two or more numbers is a number that is a multiple of each of the given numbers. So 15,000 and 30,000 are common multiples of 7,500 and 5,000.

The least common multiple (LCM) of two or more numbers is the common multiple with the least value. The LCM of 7,500 and 5,000 is 15,000. This is the lowest mileage at which both services are due at the same time.

Additional Example 1: Using a List to Find the LCM Find the least common multiple (LCM). A. 2, 7 Multiples of 2: 2, 4, 6, 8, 10, 12, 14 List some multiples of each number. Multiples of 7: 7, 14, 21, 28, 35 Find the least value that is in both lists. The LCM is 14. B. 3, 6, 9 Multiples of 3: 3, 6, 9, 12, 15, 18, 21 List some multiples of each number. Multiples of 6: 6, 12, 18, 24, 30 Multiples of 9: 9, 18, 27, 36, 45 Find the least value that is in all the lists. The LCM is 18.

Check It Out: Example 1 Find the least common multiple (LCM). A. 3, 7 Multiples of 3: 3, 6, 9, 12, 15, 18, 21 List some multiples of each number. Multiples of 7: 7, 14, 21, 28 Find the least value that is in both lists. The LCM is 21. B. 2, 6, 4 Multiples of 2: 2, 4, 6, 8, 10, 12, 14 List some multiples of each number. Multiples of 6: 6, 12, 18 Multiples of 4: 4, 8, 12 Find the least value that is in all the lists. The LCM is 12.

Additional Example 2A: Using Prime Factorization to Find the LCM Find the least common multiple (LCM). 60, 130 60 = 2 · 2 · 3 · 5 Write the prime factorization of each number. 130 = 2 · 5 · 13 Circle the common prime factors. 2, 2, 3, 5, 13 List the prime factors, using the circled factors only once. 2 · 2 · 3 · 5 · 13 Multiply the factors in the list. The LCM is 780.

Additional Example 2B: Using Prime Factorization to Find the LCM Find the least common multiple (LCM). 14, 35, 49 14 = 2 · 7 Write the prime factorization of each number. 35 = 5 · 7 Circle the common prime factors. 49 = 7 · 7 List the prime factors, using the circled factors only once. 2, 5, 7, 7 2 · 5 · 7 · 7 Multiply the factors in the list. The LCM is 490.

Find the least common multiple (LCM). Check It Out: Example 2A Find the least common multiple (LCM). 50, 110 50 = 2 · 5 · 5 Write the prime factorization of each number. 110 = 2 · 5 · 11 Circle the common prime factors. List the prime factors, using the circled factors only once. 2, 5, 5, 11 2 · 5 · 5 · 11 Multiply the factors in the list. The LCM is 550.

Check It Out: Example 2B 18, 36, 54 18 = 2 · 3 · 3 36 = 2 · 2 · 3 · 3 Write the prime factorization of each number. 36 = 2 · 2 · 3 · 3 Circle the common prime factors. 54 = 2 · 3 · 3 · 3 List the prime factors, using the circled factors only once. 2, 2, 3, 3, 3 2 · 2 · 3 · 3 · 3 Multiply the factors in the list. The LCM is 108.

Additional Example 3: Application Mr. Washington will set up the band chairs all in rows of 6 or all in rows of 8. What is the least number of chairs he will set up? Find the LCM of 6 and 8. 6 = 2 · 3 8 = 2 · 2 · 2 The LCM is 2 · 2 · 2 · 3 = 24. He will set up at least 24 chairs.

Check It Out: Example 3 Two satellites are put into orbit over the same location at the same time. One orbits the earth every 24 hours, while the second completes an orbit every 18 hours. How much time will elapse before they are once again over the same location at the same time? Find the LCM of 24 and 18. 24 = 2 · 2 · 2 · 3 18 = 2 · 3 · 3 The LCM is 2 · 2 · 2 · 3 · 3 = 72. 72 hours will elapse before they are over the same location at the same time.

Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

Find the least common multiple (LCM). 1. 18, 21 3. 4, 6, 15 Lesson Quiz Find the least common multiple (LCM). 1. 18, 21 3. 4, 6, 15 5. You are planning a picnic. You can purchase paper plates in packages of 30, paper napkins in packages of 50, and paper cups in packages of 20. What is the least number of each type of package that you can buy and have an equal number of each? 126 2. 24, 27 216 60 4. 4, 8, 16 16 o 10 packages of plates 6 packages of napkins 15 packages of cups

Lesson Quiz for Student Response Systems 1. Identify the least common multiple (LCM) of 15 and 24. A. 105 B. 112 C. 120 D. 124 o

Lesson Quiz for Student Response Systems 2. Identify the least common multiple (LCM) of 21 and 24. A. 162 B. 168 C. 172 D. 178 o

Lesson Quiz for Student Response Systems 3. Identify the least common multiple (LCM) of 6, 8, and 18. A. 72 B. 64 C. 56 D. 48 o

Lesson Quiz for Student Response Systems 4. Identify the least common multiple (LCM) of 5, 10, and 20. A. 25 B. 20 C. 15 D. 10 o

Lesson Quiz for Student Response Systems 5. You are planning to distribute pencils, notebooks, and school bags to children in an orphanage. You can purchase pencils in packages of 30, notebooks in packages of 40, and school bags in packages of 50. What is the least number of each type of package that you can buy and have an equal number of each? A. 20 packages of pencils, 15 packages of notebooks, and 12 packages of school bags B. 12 packages of pencils, 15 packages of notebooks, and 20 packages of school bags o