2-8 Least Common Multiple

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Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes

2-8 Least Common Multiple

Warm UpWrite the prime factorization of each number.

1. 68

2. 225

3. 940Find the greatest common factor.

4. 27 and 45

5. 32 and 80

22 · 17

32 · 52

22 · 5 · 47

9

16

2-8 Least Common Multiple

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

2-8 Least Common Multiple

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

2-8 Least Common Multiple

Vocabulary

multipleleast common multiple (LCM)

2-8 Least Common Multiple

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.

2-8 Least Common Multiple

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.

2-8 Least Common Multiple

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.

2-8 Least Common Multiple

Find the least common multiple (LCM).

Additional Example 1: Using a List to Find the LCM

A. 2, 7

Multiples of 2: 2, 4, 6, 8, 10, 12, 14

Multiples of 7: 7, 14, 21, 28, 35

The LCM is 14.

List some multiples of each number.

Find the least value that is in both lists.

B. 3, 6, 9

Multiples of 3: 3, 6, 9, 12, 15, 18, 21

Multiples of 6: 6, 12, 18, 24, 30

Multiples of 9: 9, 18, 27, 36, 45

The LCM is 18.

List some multiples of each number.

Find the least value that is in all the lists.

2-8 Least Common Multiple

Check It Out: Example 1

Find the least common multiple (LCM).

A. 3, 7

Multiples of 3: 3, 6, 9, 12, 15, 18, 21

Multiples of 7: 7, 14, 21, 28

The LCM is 21.

List some multiples of each number.

Find the least value that is in both lists.

B. 2, 6, 4

Multiples of 2: 2, 4, 6, 8, 10, 12, 14

Multiples of 6: 6, 12, 18

Multiples of 4: 4, 8, 12

The LCM is 12.

List some multiples of each number.

Find the least value that is in all the lists.

2-8 Least Common Multiple

Find the least common multiple (LCM).

Additional Example 2A: Using Prime Factorization to Find the LCM

60, 130

60 = 2 · 2 · 3 · 5 Write the prime factorization ofeach number.

130 = 2 · 5 · 13

2, 2, 3, 5, 13

Circle the common prime factors.

List the prime factors, usingthe circled factors only once.

2 · 2 · 3 · 5 · 13 Multiply the factors in the list.

The LCM is 780.

2-8 Least Common Multiple

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 ofeach number. 35 = 5 · 7

2, 5, 7, 7

Circle the common prime factors.

List the prime factors, usingthe circled factors only once.

2 · 5 · 7 · 7 Multiply the factors in the list.

The LCM is 490.

49 = 7 · 7

2-8 Least Common Multiple

Check It Out: Example 2A

Find the least common multiple (LCM).

50, 110

50 = 2 · 5 · 5 Write the prime factorization ofeach number.

110 = 2 · 5 · 11

2, 5, 5, 11

Circle the common prime factors.

List the prime factors, usingthe circled factors only once.

2 · 5 · 5 · 11 Multiply the factors in the list.

The LCM is 550.

2-8 Least Common Multiple

Check It Out: Example 2B

18, 36, 54

18 = 2 · 3 · 3 Write the prime factorization ofeach number.

36 = 2 · 2 · 3 · 3

2, 2, 3, 3, 3

Circle the common prime factors.

List the prime factors, usingthe circled factors only once.

2 · 2 · 3 · 3 · 3 Multiply the factors in the list.

The LCM is 108.

54 = 2 · 3 · 3 · 3

2-8 Least Common Multiple

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?

Additional Example 3: Application

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.

2-8 Least Common Multiple

Check It Out: Example 3Two 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?

24 = 2 · 2 · 2 · 3

18 = 2 · 3 · 3

The LCM is 2 · 2 · 2 · 3 · 3 = 72.

Find the LCM of 24 and 18.

72 hours will elapse before they are over the same location at the same time.

2-8 Least Common Multiple

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

2-8 Least Common Multiple

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?

216126

60 16

2. 24, 27

4. 4, 8, 16

10 packages of plates6 packages of napkins15 packages of cups

2-8 Least Common Multiple

1. Identify the least common multiple (LCM) of 15 and 24.

A. 105

B. 112

C. 120

D. 124

Lesson Quiz for Student Response Systems

2-8 Least Common Multiple

2. Identify the least common multiple (LCM) of 21 and 24.

A. 162

B. 168

C. 172

D. 178

Lesson Quiz for Student Response Systems

2-8 Least Common Multiple

3. Identify the least common multiple (LCM) of 6, 8, and 18.

A. 72

B. 64

C. 56

D. 48

Lesson Quiz for Student Response Systems

2-8 Least Common Multiple

4. Identify the least common multiple (LCM) of 5, 10, and 20.

A. 25

B. 20

C. 15

D. 10

Lesson Quiz for Student Response Systems

2-8 Least Common Multiple

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

Lesson Quiz for Student Response Systems