Добавил:
Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз: Предмет: Файл:
basic mathematics for college students.pdf
Скачиваний:
147
Добавлен:
10.06.2015
Размер:
15.55 Mб
Скачать

1.8

The prime factorization of a number is given. What is the number? See Example 9.

93.

2

3 3

5

94.

2

2 2 7

95.

7

112

 

96.

2

34

97.

32

52

 

98.

33 53

99.

23

33

13

100.

23 32 11

APPLICATIONS

101. PERFECT NUMBERS A whole number is called a perfect number when the sum of its factors that are less than the number equals the number. For example, 6 is a perfect number,

because 1 2 3 6. Find the factors of 28. Then use addition to show that 28 is also a perfect number.

102. CRYPTOGRAPHY Information is often transmitted in code. Many codes involve writing products of large primes, because they are difficult to factor. To see how difficult, try finding two prime factors of 7,663. (Hint: Both primes are greater than 70.)

103. LIGHT The illustration shows that the light energy that passes through the first unit of area, 1 yard away from the bulb, spreads out as it travels away from the source. How much area does that energy cover 2 yards, 3 yards, and 4 yards from the bulb? Express each answer using exponents.

1 square unit

1 yd 2 yd

3 yd

4 yd

The Least Common Multiple and the Greatest Common Factor

89

104.CELL DIVISION After 1 hour, a cell has divided to form another cell. In another hour, these two cells have divided so that four cells exist. In another hour, these four cells divide so that eight exist.

a.How many cells exist at the end of the fourth hour?

b.The number of cells that exist after each division can be found using an exponential expression. What is the base?

c.Find the number of cells after 12 hours.

WRITING

105.Explain how to check a prime factorization.

106.Explain the difference between the factors of a number and the prime factors of a number. Give an example.

107.Find 12, 13, and 14. From the results, what can be said about any power of 1?

108.Use the phrase infinitely many in a sentence.

REVIEW

109.MARCHING BANDS When a university band lines up in eight rows of fifteen musicians, there are five musicians left over. How many band members are there?

110.U.S. COLLEGE COSTS In 2008, the average yearly tuition cost and fees at a private four-year college was $25,143. The average yearly tuition cost and fees at a public four-year college was $6,585. At these rates, how much less are the tuition costs and fees at a public college over four years? (Source: The College Board)

S E C T I O N 1.8

The Least Common Multiple and the Greatest

Common Factor

As a child, you probably learned how to count by 2’s and 5’s and 10’s. Counting in that way is an example of an important concept in mathematics called multiples.

1 Find the LCM by listing multiples.

Objectives

1Find the LCM by listing multiples.

2Find the LCM using prime factorization.

3Find the GCF by listing factors.

4Find the GCF using prime factorization.

The multiples of a number are the products of that number and 1, 2, 3, 4, 5, and so on.

90

Chapter 1 Whole Numbers

Self Check 1

Find the first eight multiples of 9.

Now Try Problems 17 and 85

EXAMPLE 1 Find the first eight multiples of 6.

Strategy We will multiply 6 by 1, 2, 3, 4, 5, 6, 7, and 8.

WHY The multiples of a number are the products of that number and 1, 2, 3, 4, 5, and so on.

Solution

To find the multiples, we proceed as follows:

6 1 6 This is the first multiple of 6.

6 2 12

6 3 18

6 4 24

6 5 30

6 6 36

6 7 42

6 8 48 This is the eighth multiple of 6.

The first eight multiples of 6 are 6, 12, 18, 24, 30, 36, 42, and 48.

The first eight multiples of 3 and the first eight multiples of 4 are shown below. The numbers highlighted in red are common multiples of 3 and 4.

3

1 3

4

1 4

3

2 6

4

2 8

3

3 9

4

3 12

3

4 12

4

4 16

3

5 15

4

5 20

3

6 18

4

6 24

3

7 21

4

7 28

3

8 24

4

8 32

If we extend each list, it soon becomes apparent that 3 and 4 have infinitely many common multiples.

The common multiples of 3 and 4 are: 12, 24, 36, 48, 60, 72, p

Because 12 is the smallest number that is a multiple of both 3 and 4, it is called the least common multiple (LCM) of 3 and 4. We can write this in compact form as:

LCM (3, 4) 12 Read as “The least common multiple of 3 and 4 is 12.”

The Least Common Multiple (LCM)

The least common multiple of two whole numbers is the smallest common multiple of the numbers.

We have seen that the LCM of 3 and 4 is 12. It is important to note that 12 is divisible by both 3 and 4.

12

4 and

12

3

3

4

This observation illustrates an important relationship between divisibility and the least common multiple.

1.8 The Least Common Multiple and the Greatest Common Factor

91

The Least Common Multiple (LCM)

The least common multiple (LCM) of two whole numbers is the smallest whole number that is divisible by both of those numbers.

When finding the LCM of two numbers, writing both lists of multiples can be tiresome. From the previous definition of LCM, it follows that we need only list the multiples of the larger number. The LCM is simply the first multiple of the larger number that is divisible by the smaller number. For example, to find the LCM of 3 and 4, we observe that

The multiples of 4 are:

4,

8,

12,

16, 20, 24, p

Recall that one

 

 

 

 

 

number is divisible by

 

 

 

 

 

 

 

 

 

 

another if, when

4 is not

8 is not

12 is

dividing them, we get

divisible by 3.

divisible by 3.

divisible by 3.

a remainder of 0.

Since 12 is the first multiple of 4 that is divisible by 3, the LCM of 3 and 4 is 12.

As expected, this is the same result that we obtained using the two-list method.

Finding the LCM by Listing the Multiples of the Largest Number

To find the least common multiple of two (or more) whole numbers:

1.Write multiples of the largest number by multiplying it by 1, 2, 3, 4, 5, and so on.

2.Continue this process until you find the first multiple of the larger number that is divisible by each of the smaller numbers. That multiple is their

LCM.

EXAMPLE 2 Find the LCM of 6 and 8.

Strategy We will write the multiples of the larger number, 8, until we find one that is divisible by the smaller number, 6.

WHY The LCM of 6 and 8 is the smallest multiple of 8 that is divisible by 6.

Solution

The 1st multiple of 8:

8 1

8

 

 

8 is not divisible by 6. (When we divide, we get

 

 

 

 

 

a remainder of 2.) Since 8 is not divisible by

 

 

 

 

 

6, find the next multiple.

The 2nd multiple of 8:

8 2

16

 

 

16 is not divisible by 6. Find the next

 

 

 

 

 

multiple.

The 3rd multiple of 8:

8 3

24

 

 

24 is divisible by 6. This is the LCM.

The first multiple of 8 that is divisible by 6 is 24. Thus,

LCM (6, 8) 24 Read as “The least common multiple of 6 and 8 is 24.”

We can extend this method to find the LCM of three whole numbers.

EXAMPLE 3 Find the LCM of 2, 3, and 10.

Strategy We will write the multiples of the largest number, 10, until we find one that is divisible by both of the smaller numbers, 2 and 3.

WHY The LCM of 2, 3, and 10 is the smallest multiple of 10 that is divisible by 2 and 3.

Self Check 2

Find the LCM of 8 and 10.

Now Try Problem 25

Self Check 3

Find the LCM of 3, 4, and 8.

Now Try Problem 35

Read as “The least common multiple of 2, 3, and 10 is 30.”

92

Chapter 1 Whole Numbers

Solution

The 1st multiple of 10: 10 1 10

10 is divisible by 2, but not by 3. Find the next multiple.

The 2nd multiple of 10: 10 2 20

20 is divisible by 2, but not by 3. Find the next multiple.

The 3rd multiple of 10: 10 3 30 The first multiple of 10 that is divisible by

LCM (2, 3, 10) 30

 

 

30 is divisible by 2 and by 3. It is the LCM.

2 and 3 is 30. Thus,

2 Find the LCM using prime factorization.

Another method for finding the LCM of two (or more) whole numbers uses prime factorization. This method is especially helpful when working with larger numbers. As an example, we will find the LCM of 36 and 54. First, we find their prime factorizations:

36 2 2 3 3

Factor trees (or division

 

36

 

 

 

54

 

 

ladders) can be used to

4

 

9

 

6

 

9

 

find the prime factorizations.

 

 

 

 

 

 

 

 

 

 

 

54 2 3 3 3

2

2

3

3

2

3

3

3

 

 

 

 

 

 

 

 

The LCM of 36 and 54 must be divisible by 36 and 54. If the LCM is divisible by 36, it must have the prime factors of 36, which are 2 2 3 3. If the LCM is divisible by 54, it must have the prime factors of 54, which are 2 3 3 3. The smallest number that meets both requirements is

These are the prime factors of 36.

 

 

 

 

2 2 3 3 3

 

 

 

 

These are the prime factors of 54.

To find the LCM, we perform the indicated multiplication:

LCM (36, 54) 2 2 3 3 3 108

Caution! The LCM (36, 54) is not the product of the prime factorization of 36 and the prime factorization of 54. That gives an incorrect answer of 2,052.

LCM (36, 54) 2 2 3 3 2 3 3 3 1,944

The LCM should contain all the prime factors of 36 and all the prime factors of 54, but the prime factors that 36 and 54 have in common are not repeated.

The prime factorizations of 36 and 54 contain the numbers 2 and 3.

36 2 2 3 3

54 2 3 3 3

We see that

The greatest number of times the factor 2 appears in any one of the prime factorizations is twice and the LCM of 36 and 54 has 2 as a factor twice.

The greatest number of times that 3 appears in any one of the prime factorizations is three times and the LCM of 36 and 54 has 3 as a factor three times.

These observations suggest a procedure to use to find the LCM of two (or more) numbers using prime factorization.

1.8 The Least Common Multiple and the Greatest Common Factor

93

Finding the LCM Using Prime Factorization

To find the least common multiple of two (or more) whole numbers:

1.Prime factor each number.

2.The LCM is a product of prime factors, where each factor is used the greatest number of times it appears in any one factorization.

EXAMPLE 4 Find the LCM of 24 and 60.

Strategy We will begin by finding the prime factorizations of 24 and 60.

WHY To find the LCM, we need to determine the greatest number of times each prime factor appears in any one factorization.

Self Check 4

Find the LCM of 18 and 32.

Now Try Problem 37

Solution

Step 1 Prime factor 24 and 60.

2

24

2

60

 

 

24 2 2 2 3

Division ladders (or factor trees) can be

2

 

 

12

2

 

30

 

 

 

 

 

 

2

 

6

3

 

15

 

 

 

 

 

60 2 2 3 5

used to find the prime factorizations.

 

3

 

 

5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Step 2 The prime factorizations of 24 and 60 contain the prime factors 2, 3, and 5. To find the LCM, we use each of these factors the greatest number of times it appears in any one factorization.

We will use the factor 2 three times, because 2 appears three times in the factorization of 24. Circle 2 2 2, as shown below.

We will use the factor 3 once, because it appears one time in the factorization of 24 and one time in the factorization of 60. When the number of times a factor appears are equal, circle either one, but not both, as shown below.

We will use the factor 5 once, because it appears one time in the factorization of 60. Circle the 5, as shown below.

24 2 2 2 3

60 2 2 3 5

Since there are no other prime factors in either prime factorization, we have

 

 

 

 

 

 

 

 

 

Use 2 three times.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Use 3 one time.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Use 5 one time.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

⎪ ⎬

3 5 120

 

LCM (24, 60) 2

2

2

 

Note that 120 is the smallest number that is divisible by both 24 and 60:

120

5 and

120

2

 

24

60

 

 

In Example 4, we can express the prime factorizations of 24 and 60 using exponents. To determine the greatest number of times each factor appears in any one factorization, we circle the factor with the greatest exponent.

94

Chapter 1 Whole Numbers

Self Check 5

Find the LCM of 45, 60, and 75.

Now Try Problem 45

24

23

 

31

The greatest exponent on the factor 2 is 3.

 

 

 

 

The greatest exponent on the factor 3 is 1.

60

22

 

31 51

The greatest exponent on the factor 5 is 1.

The LCM of 24 and 60 is

 

23 31 51 8 3 5 120

Evaluate: 23 8.

EXAMPLE 5

Find the LCM of 28, 42, and 45.

 

Strategy We will begin by finding the prime factorizations of 28, 42, and 45.

WHY To find the LCM, we need to determine the greatest number of times each prime factor appears in any one factorization.

Solution

Step 1 Prime factor 28, 42, and 45.

28 2 2 7 This can be written as 22 71.

42 2 3 7 This can be written as 21 31 71 .

45 3 3 5 This can be written as 32 5 .

Step 2 The prime factorizations of 28, 42, and 45 contain the prime factors 2, 3, 5, and 7. To find the LCM (28, 42, 45), we use each of these factors the greatest number of times it appears in any one factorization.

We will use the factor 2 two times, because 2 appears two times in the factorization of 28. Circle 2 2, as shown above.

We will use the factor 3 twice, because it appears two times in the factorization of 45. Circle 3 3, as shown above.

We will use the factor 5 once, because it appears one time in the factorization of 45. Circle the 5, as shown above.

We will use the factor 7 once, because it appears one time in the factorization of 28 and one time in the factorization of 42. You may circle either 7, but only circle one of them.

Since there are no other prime factors in either prime factorization, we have

Use the factor 2 two times.

Use the factor 3 two times.

Use the factor 5 one time.

Use the factor 7 one time.

 

 

 

 

 

 

LCM (28, 42, 45)

⎬ ⎭ ⎫ ⎬ ⎭

5

7 1,260

2 2

3

3

If we use exponents, we have

 

LCM (28, 42, 45)

22 32

5 7

1,260

Either way, we have found that the LCM (28, 42, 45) 1,260. Note that 1,260 is the smallest number that is divisible by 28, 42, and 45:

1,260

315

1,260

30

1,260

28

 

4

42

45

 

 

EXAMPLE 6

Patient Recovery Two patients recovering from heart

 

surgery exercise daily by walking around a track. One patient can complete a lap in 4 minutes. The other can complete a lap in 6 minutes. If they begin at the same time and at the same place on the track, in how many minutes will they arrive together at the starting point of their workout?

EXAMPLE 7

1.8

The Least Common Multiple and the Greatest Common Factor

95

Strategy We will find the LCM of 4 and 6.

 

Self Check 6

 

 

 

WHY Since one patient reaches the starting point of the workout every 4 minutes, and the other is there every 6 minutes, we want to find the least common multiple of those numbers. At that time, they will both be at the starting point of the workout.

Solution

To find the LCM, we prime factor 4 and 6, and circle each prime factor the greatest number of times it appears in any one factorization.

4 2 2 Use the factor 2 two times, because 2 appears two times in the factorization of 4.

AQUARIUMS A pet store owner changes the water in a fish aquarium every 45 days and he changes the pump filter every

20 days. If the water and filter are changed on the same day, in how many days will they be changed again together?

Now Try Problem 87

6 2 3 Use the factor 3 once, because it appears one time in the factorization of 6.

Since there are no other prime factors in either prime factorization, we have

LCM (4, 6) 2 2 3 12

The patients will arrive together at the starting point 12 minutes after beginning their workout.

3 Find the GCF by listing factors.

We have seen that two whole numbers can have common multiples. They can also have common factors. To explore this concept, let’s find the factors of 26 and 39 and see what factors they have in common.

To find the factors of 26, we find all the pairs of whole numbers whose product is 26. There are two possibilities:

1 26 26 2 13 26

Each of the numbers in the pairs is a factor of 26. From least to greatest, the factors of 26 are 1, 2, 13, and 26.

To find the factors of 39, we find all the pairs of whole numbers whose product is 39. There are two possibilities:

1 39 39 3 13 39

Each of the numbers in the pairs is a factor of 39. From least to greatest, the factors of 39 are 1, 3, 13, and 39. As shown below, the common factors of 26 and 39 are 1 and 13.

1 , 2 , 13 , 26 These are the factors of 26.

1 , 3 , 13 , 39 These are the factors of 39.

Because 13 is the largest number that is a factor of both 26 and 39, it is called the greatest common factor (GCF) of 26 and 39. We can write this in compact form as:

GCF (26, 39) 13 Read as “The greatest common factor of 26 and 39 is 13.”

The Greatest Common Factor (GCF)

The greatest common factor of two whole numbers is the largest common factor of the numbers.

Find the GCF of 18 and 45.

Strategy We will find the factors of 18 and 45.

WHY Then we can identify the largest factor that 18 and 45 have in common.

Self Check 7

Find the GCF of 30 and 42.

Now Try Problem 49

96

Chapter 1 Whole Numbers

Self Check 8

Find the GCF of 36 and 60.

Now Try Problem 57

Solution

To find the factors of 18, we find all the pairs of whole numbers whose product is 18. There are three possibilities:

1 18 18 2 9 18 3 6 18

To find the factors of 45, we find all the pairs of whole numbers whose product is 45. There are three possibilities:

1 45 45 3 15 45 5 9 45

The factors of 18 and 45 are listed below. Their common factors are circled.

Factors of 18:

1 ,

2,

3 ,

6,

9

,

18

Factors of 45:

1 ,

3 ,

5 ,

9 ,

15

,

45

 

 

 

 

 

 

 

 

The common factors of 18 and 45 are 1, 3, and 9. Since 9 is their largest common factor,

GCF (18, 45) 9 Read as “The greatest common factor of 18 and 45 is 9.”

In Example 7, we found that the GCF of 18 and 45 is 9. Note that 9 is the greatest number that divides 18 and 45.

18

2

45

5

9

9

In general, the greatest common factor of two (or more) numbers is the largest number that divides them exactly. For this reason, the greatest common factor is also known as the greatest common divisor (GCD) and we can write GCD (18, 45) 9.

4 Find the GCF using prime factorization.

We can find the GCF of two (or more) numbers by listing the factors of each number. However, this method can be lengthy. Another way to find the GCF uses the prime factorization of each number.

Finding the GCF Using Prime Factorization

To find the greatest common factor of two (or more) whole numbers:

1.Prime factor each number.

2.Identify the common prime factors.

3.The GCF is a product of all the common prime factors found in Step 2. If there are no common prime factors, the GCF is 1.

EXAMPLE 8 Find the GCF of 48 and 72.

Strategy We will begin by finding the prime factorizations of 48 and 72.

WHY Then we can identify any prime factors that they have in common.

Solution

Step 1 Prime factor 48 and 72.

48

 

 

72

 

 

 

 

 

 

48 2 2 2 2 3

4

12

9

8

 

 

 

 

 

 

72 2 2 2 3 3

2 2 4

3

3 3 2 4

 

 

2

2

 

2

2

EXAMPLE 11

1.8 The Least Common Multiple and the Greatest Common Factor

97

Step 2 The circling on the previous page shows that 48 and 72 have four common prime factors: Three common factors of 2 and one common factor of 3.

Step 3 The GCF is the product of the circled prime factors.

GCF (48, 72) 2 2 2 3 24

EXAMPLE 9 Find the GCF of 8 and 15.

Strategy We will begin by finding the prime factorizations of 8 and 15.

WHY Then we can identify any prime factors that they have in common.

Solution

The prime factorizations of 8 and 15 are shown below.

8 2 2 2

15 3 5

Since there are no common factors, the GCF of 8 and 15 is 1. Thus,

GCF (8, 15) 1 Read as “The greatest common factor of 8 and 15 is 1.”

Self Check 9

Find the GCF of 8 and 25.

Now Try Problem 61

EXAMPLE 10 Find the GCF of 20, 60, and 140.

Strategy We will begin by finding the prime factorizations of 20, 60, and 140.

WHY Then we can identify any prime factors that they have in common.

Solution

The prime factorizations of 20, 60, and 140 are shown below.

20 2 2 5

60 2 2 3 5

140 2 2 5 7

The circling above shows that 20, 60, and 140 have three common factors: two common factors of 2 and one common factor of 5. The GCF is the product of the circled prime factors.

GCF (20, 60, 140) 2 2 5 20 Read as “The greatest common factor of 20, 60, and 140 is 20.”

Note that 20 is the greatest number that divides 20, 60, and 140 exactly.

20

1

60

3

140

7

 

20

20

20

 

 

Bouquets A florist wants to use 12 white tulips, 30 pink tulips, and 42 purple tulips to make as many identical arrangements as possible. Each bouquet is to have the same number of each color tulip.

a.What is the greatest number of arrangements that she can make?

b.How many of each type of tulip can she use in each bouquet? Strategy We will find the GCF of 12, 30, and 42.

WHY Since an equal number of tulips of each color will be used to create the identical arrangements, division is indicated. The greatest common factor of three numbers is the largest number that divides them exactly.

Self Check 10

Find the GCF of 45, 60, and 75.

Now Try Problem 67

Self Check 11

SCHOOL SUPPLIES A bookstore manager wants to use some leftover items (36 markers,

54 pencils, and 108 pens) to make identical gift packs to donate to an elementary school.

a.What is the greatest number of gift packs that can be made? (continued)

98

Chapter 1 Whole Numbers

b.How many of each type of item will be in each gift pack?

Now Try Problem 93

Solution

a.To find the GCF, we prime factor 12, 30, and 42, and circle the prime factors that they have in common.

12 2

2

3

30 2

3

5

42 2

3

7

The GCF is the product of the circled numbers.

GCF (12, 30, 42) 2 3 6

The florist can make 6 identical arrangements from the tulips.

b.To find the number of white, pink, and purple tulips in each of the 6 arrangements, we divide the number of tulips of each color by 6.

White tulips:

Pink tulips:

Purple tulips:

12

2

30

5

42

7

 

 

 

 

 

 

6

6

6

Each of the 6 identical arrangements will contain 2 white tulips, 5 pink tulips, and 7 purple tulips.

 

ANSWERS TO SELF CHECKS

 

 

 

 

1.

9, 18, 27, 36, 45, 54, 63, 72 2. 40

3. 24 4. 288 5. 900 6. 180 days 7. 6 8. 12

 

9.

1 10. 15 11. a. 18 gift packs

b. 2 markers, 3 pencils, 6 pens

 

 

 

 

 

 

S E C T I O N

1.8 STUDY SET

 

 

 

VOCABULARY

 

 

 

 

 

 

 

 

 

Fill in the blanks.

 

 

 

 

 

1. The

 

 

of a number are the products of that

number and 1, 2, 3, 4, 5, and so on.

2. Because 12 is the smallest number that is a multiple of both 3 and 4, it is the of

3 and 4.

3. One number is by another if, when dividing

them, we get a remainder of 0.

4. Because 6 is the largest number that is a factor of both 18 and 24, it is the of

18 and 24.

CONCEPTS

5.a. The LCM of 4 and 6 is 12. What is the smallest whole number divisible by 4 and 6?

b. Fill in the blank: In general, the LCM of two whole numbers is the whole number that is divisible by both numbers.

6.a. What are the common multiples of 2 and 3 that appear in the list of multiples shown in the next column?

b. What is the LCM of 2 and 3?

Multiples of 2

Multiples of 3

2

1 2

3

1 3

2

2 4

3

2 6

2

3

6

3

3

9

2

4

8

3

4

12

2

5

10

3

5

15

2

6

12

3

6

18

7.a. The first six multiples of 5 are 5, 10, 15, 20, 25, and 30. What is the first multiple of 5 that is divisible by 4?

b. What is the LCM of 4 and 5?

8.Fill in the blanks to complete the prime factorization of 24.

24

4

2

9. The prime factorizations of 36 and 90 are:

36 2 2 3 3

90 2 3 3 5

What is the greatest number of times

a.2 appears in any one factorization?

b.3 appears in any one factorization?

c.5 appears in any one factorization?

d.Fill in the blanks to find the LCM of 36 and 90:

LCM

10. The prime factorizations of 14, 70, and 140 are:

14

2

7

 

70

2

5

7

140

2

2

5 7

What is the greatest number of times

a.2 appears in any one factorization?

b.5 appears in any one factorization?

c.7 appears in any one factorization?

d.Fill in the blanks to find the LCM of 14, 70, and 140:

LCM

11. The prime factorizations of 12 and 54 are:

12 22 31

54 21 33

What is the greatest number of times

a.2 appears in any one factorization?

b.3 appears in any one factorization?

c.Fill in the blanks to find the LCM of 12 and 54:

LCM 2 3

12. The factors of 18 and 45 are shown below.

Factors of 18:

1, 2, 3, 6,

9, 18

Factors of 45:

1, 3, 5, 9,

15, 45

a.Circle the common factors of 18 and 45.

b.What is the GCF of 18 and 45?

13.The prime factorizations of 60 and 90 are:

60 2 2 3 5

90 2 3 3 5

a.Circle the common prime factors of 60 and 90.

b.What is the GCF of 60 and 90?

14.The prime factorizations of 36, 84, and 132 are:

36 2 2 3 3

84 2 2 3 7

132 2 2 3 11

a.Circle the common factors of 36, 84, and 132.

b.What is the GCF of 36, 84, and 132?

1.8 The Least Common Multiple and the Greatest Common Factor

99

 

 

NOTATION

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

15.

a. The abbreviation for the greatest common factor

 

 

 

 

 

is

 

.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b. The abbreviation for the least common multiple is

 

 

 

 

 

.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

16.

a. We read LCM (2, 15) 30 as “The

 

 

 

 

 

 

 

 

 

 

 

 

multiple

 

 

 

2 and 15

 

 

 

 

30.”

 

 

 

 

b. We read GCF (18, 24) 6 as “The

 

 

 

 

 

 

 

 

 

 

 

factor

 

 

18 and 24

 

6.”

 

 

 

GUIDED PRACTICE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Find the first eight multiples of each number. See Example 1.

 

17.

4

 

 

 

 

 

 

 

18.

2

 

 

 

 

 

 

 

 

 

19.

11

 

 

 

 

 

 

 

20.

10

 

 

 

 

 

 

 

 

 

21.

8

 

 

 

 

 

 

 

22.

9

 

 

 

 

 

 

 

 

 

23.

20

 

 

 

 

 

 

 

24.

30

 

 

 

 

 

 

 

 

 

 

Find the LCM of the given numbers. See Example 2.

 

25.

3, 5

 

 

 

 

 

 

26.

6, 9

 

 

 

 

 

 

 

 

 

27.

8, 12

 

 

 

 

 

 

28.

10, 25

 

 

 

 

 

 

 

 

29.

5, 11

 

 

 

 

 

 

30.

7, 11

 

 

 

 

 

 

 

 

 

31.

4, 7

 

 

 

 

 

 

32.

5, 8

 

 

 

 

 

 

 

 

 

 

Find the LCM of the given numbers. See Example 3.

 

33.

3, 4, 6

 

 

 

 

 

34.

2, 3, 8

 

 

 

 

 

 

 

 

35.

2, 3, 10

 

 

 

 

 

36.

3, 6, 15

 

 

 

 

 

 

 

 

 

Find the LCM of the given numbers. See Example 4.

 

37.

16, 20

 

 

 

 

 

38.

14, 21

 

 

 

 

 

 

 

 

39.

30, 50

 

 

 

 

 

40.

21, 27

 

 

 

 

 

 

 

 

41.

35, 45

 

 

 

 

 

42.

36, 48

 

 

 

 

 

 

 

 

43.

100, 120

 

 

 

 

 

44.

120, 180

 

 

 

 

 

 

 

 

Find the LCM of the given numbers. See Example 5.

 

45.

6, 24, 36

 

 

 

 

 

46.

6, 10, 18

 

 

 

 

 

 

 

47.

5, 12, 15

 

 

 

 

 

48.

8, 12, 16

 

 

 

 

 

 

 

 

Find the GCF of the given numbers. See Example 7.

 

49.

4, 6

 

 

 

 

 

 

50.

6, 15

 

 

 

 

 

 

 

 

 

51.

9, 12

 

 

 

 

 

 

52.

10, 12

 

 

 

 

 

 

 

 

70. 120, 180
72. 15, 300
74. 52, 78, 130
76. 9, 16, 25
78. 98, 102
80. 26, 39, 65
82. 38, 57
84. 65, 81

100

Chapter 1

Whole Numbers

Find the GCF of the given numbers. See Example 8.

53.

22, 33

54.

14, 21

55.

15, 30

56.

15, 75

57.

18, 96

58.

30, 48

59.

28, 42

60.

63, 84

Find the GCF of the given numbers. See Example 9.

61.

16, 51

62.

27, 64

63.

81, 125

64.

57, 125

Find the GCF of the given numbers. See Example 10.

65.

12, 68, 92

66.

24, 36, 40

67.

72, 108, 144

68.

81, 108, 162

TRY IT YOURSELF

Find the LCM and the GCF of the given numbers.

69. 100, 120

71. 14, 140

73. 66, 198, 242

75. 8, 9, 49

77. 120, 125

79. 34, 68, 102

81. 46, 69

83. 50, 81

APPLICATIONS

85.OIL CHANGES Ford has officially extended the oil change interval for 2007 and newer cars to every 7,500 miles. (It used to be every 5,000 miles). Complete the table below that shows Ford’s new recommended oil change mileages.

1st oil

2nd oil

3rd oil

4th oil

5th oil

6th oil

change

change

change

change

change

change

 

 

 

 

 

 

7,500 mi

 

 

 

 

 

 

 

 

 

 

 

86.ATMs An ATM machine offers the customer cash withdrawal choices in multiples of $20.

The minimum withdrawal is $20 and the maximum is $200. List the dollar amounts

of cash that can be withdrawn from the ATM machine.

87.NURSING A nurse is instructed to check a patient’s blood pressure every 45 minutes and another is instructed to take the same patient’s temperature every 60 minutes. If both nurses are in the patient’s room now, how long will it be until the nurses are together in the room once again?

88.BIORHYTHMS Some scientists believe that there are natural rhythms of the body, called biorhythms, that affect our physical, emotional, and mental cycles. Our physical biorhythm cycle lasts 23 days, the emotional biorhythm cycle lasts 28 days, and our mental biorhythm cycle lasts 33 days. Each biorhythm cycle has a high, low and critical zone. If your three cycles are together one day, all at their lowest point, in how many more days will they be together again, all at their lowest point?

89.PICNICS A package of hot dogs usually contains 10 hot dogs and a package of buns usually contains 12 buns. How many packages of hot dogs and buns should a person buy to be sure that there are equal numbers of each?

90.WORKING COUPLES A husband works for

6 straight days and then has a day off. His wife works for 7 straight days and then has a day off. If the husband and wife are both off from work on the same day, in how many days will they both be off from work again?

91.DANCE FLOORS A dance floor is to be made from rectangular pieces of plywood that are 6 feet by

8 feet. What is the minimum number of pieces of plywood that are needed to make a square dance floor?

6 ft

8 ft

Plywood

sheet

Square dance floor

92.BOWLS OF SOUP Each of the bowls shown below holds an exact number of full ladles of soup.

a.If there is no spillage, what is the greatest-size ladle (in ounces) that a chef can use to fill all three bowls?

b.How many ladles will it take to fill each bowl?

12 ounces

21 ounces

18 ounces