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Prealgebra 2e

2.4 Find Multiples and Factors

Prealgebra 2e2.4 Find Multiples and Factors
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

Learning Objectives

By the end of this section, you will be able to:
  • Identify multiples of numbers
  • Use common divisibility tests
  • Find all the factors of a number
  • Identify prime and composite numbers
Be Prepared 2.10

Before you get started, take this readiness quiz.

Which of the following numbers are counting numbers (natural numbers)?
0,4,2150,4,215
If you missed this problem, review Example 1.1.

Be Prepared 2.11

Find the sum of 3,5,3,5, and 7.7.
If you missed the problem, review Example 2.1.

Identify Multiples of Numbers

Annie is counting the shoes in her closet. The shoes are matched in pairs, so she doesn’t have to count each one. She counts by twos: 2,4,6,8,10,12.2,4,6,8,10,12. She has 1212 shoes in her closet.

The numbers 2,4,6,8,10,122,4,6,8,10,12 are called multiples of 2.2. Multiples of 22 can be written as the product of a counting number and 2.2. The first six multiples of 22 are given below.

12=222=432=642=852=1062=1212=222=432=642=852=1062=12

A multiple of a number is the product of the number and a counting number. So a multiple of 33 would be the product of a counting number and 3.3. Below are the first six multiples of 3.3.

13=323=633=943=1253=1563=1813=323=633=943=1253=1563=18

We can find the multiples of any number by continuing this process. Table 2.8 shows the multiples of 22 through 99 for the first twelve counting numbers.

Counting Number 11 22 33 44 55 66 77 88 99 1010 1111 1212
Multiples of2Multiples of2 22 44 66 88 1010 1212 1414 1616 1818 2020 2222 2424
Multiples of3Multiples of3 33 66 99 1212 1515 1818 2121 2424 2727 3030 3333 3636
Multiples of4Multiples of4 44 88 1212 1616 2020 2424 2828 3232 3636 4040 4444 4848
Multiples of5Multiples of5 55 1010 1515 2020 2525 3030 3535 4040 4545 5050 5555 6060
Multiples of6Multiples of6 66 1212 1818 2424 3030 3636 4242 4848 5454 6060 6666 7272
Multiples of7Multiples of7 77 1414 2121 2828 3535 4242 4949 5656 6363 7070 7777 8484
Multiples of8Multiples of8 88 1616 2424 3232 4040 4848 5656 6464 7272 8080 8888 9696
Multiples of9Multiples of9 99 1818 2727 3636 4545 5454 6363 7272 8181 9090 9999 108108
Table 2.8

Multiple of a Number

A number is a multiple of nn if it is the product of a counting number and n.n.

Recognizing the patterns for multiples of 2,5,10,and32,5,10,and3 will be helpful to you as you continue in this course.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Multiples” will help you develop a better understanding of multiples.

Figure 2.6 shows the counting numbers from 11 to 50.50. Multiples of 22 are highlighted. Do you notice a pattern?

The image shows a chart with five rows and ten columns. The first row lists the numbers from 1 to 10. The second row lists the numbers from 11 to 20. The third row lists the numbers from 21 to 30. The fourth row lists the numbers from 31 and 40. The fifth row lists the numbers from 41 to 50. All factors of 2 are highlighted in blue.
Figure 2.6 Multiples of 22 between 11 and 5050

The last digit of each highlighted number in Figure 2.6 is either 0,2,4,6,or8.0,2,4,6,or8. This is true for the product of 22 and any counting number. So, to tell if any number is a multiple of 22 look at the last digit. If it is 0,2,4,6,or8,0,2,4,6,or8, then the number is a multiple of 2.2.

Example 2.40

Determine whether each of the following is a multiple of 2:2:

  1. 489489
  2. 3,7143,714
Try It 2.79

Determine whether each number is a multiple of 2:2:

  1. 678678
  2. 21,49321,493
Try It 2.80

Determine whether each number is a multiple of 2:2:

  1. 979 979
  2. 17,78017,780

Now let’s look at multiples of 5.5. Figure 2.7 highlights all of the multiples of 55 between 11 and 50.50. What do you notice about the multiples of 5?5?

The image shows a chart with five rows and ten columns. The first row lists the numbers from 1 to 10. The second row lists the numbers from 11 to 20. The third row lists the numbers from 21 to 30. The fourth row lists the numbers from 31 and 40. The fifth row lists the numbers from 41 to 50. All factors of 5 are highlighted in blue.
Figure 2.7 Multiples of 55 between 11 and 5050

All multiples of 55 end with either 55 or 0.0. Just like we identify multiples of 22 by looking at the last digit, we can identify multiples of 55 by looking at the last digit.

Example 2.41

Determine whether each of the following is a multiple of 5:5:

  1. 579 579
  2. 880880
Try It 2.81

Determine whether each number is a multiple of 5.5.

  1. 675675
  2. 1,5781,578
Try It 2.82

Determine whether each number is a multiple of 5.5.

  1. 421 421
  2. 2,6902,690

Figure 2.8 highlights the multiples of 1010 between 11 and 50.50. All multiples of 1010 all end with a zero.

The image shows a chart with five rows and ten columns. The first row lists the numbers from 1 to 10. The second row lists the numbers from 11 to 20. The third row lists the numbers from 21 to 30. The fourth row lists the numbers from 31 and 40. The fifth row lists the numbers from 41 to 50. All factors of 10 are highlighted in blue.
Figure 2.8 Multiples of 1010 between 11 and 5050

Example 2.42

Determine whether each of the following is a multiple of 10:10:

  1. 425425
  2. 350350
Try It 2.83

Determine whether each number is a multiple of 10:10:

  1. 179179
  2. 3,5403,540
Try It 2.84

Determine whether each number is a multiple of 10:10:

  1. 110 110
  2. 7,5957,595

Figure 2.9 highlights multiples of 3.3. The pattern for multiples of 33 is not as obvious as the patterns for multiples of 2,5,and10.2,5,and10.

The image shows a chart with five rows and ten columns. The first row lists the numbers from 1 to 10. The second row lists the numbers from 11 to 20. The third row lists the numbers from 21 to 30. The fourth row lists the numbers from 31 and 40. The fifth row lists the numbers from 41 to 50. All factors of 3 are highlighted in blue.
Figure 2.9 Multiples of 33 between 11 and 5050

Unlike the other patterns we’ve examined so far, this pattern does not involve the last digit. The pattern for multiples of 33 is based on the sum of the digits. If the sum of the digits of a number is a multiple of 3,3, then the number itself is a multiple of 3.3. See Table 2.9.

Multiple of 3Multiple of 3 33 66 99 1212 1515 1818 2121 2424
Sum of digitsSum of digits 33 66 99 1+231+23 1+561+56 1+891+89 2+132+13 2+462+46
Table 2.9

Consider the number 42.42. The digits are 44 and 2,2, and their sum is 4+2=6.4+2=6. Since 66 is a multiple of 3,3, we know that 4242 is also a multiple of 3.3.

Example 2.43

Determine whether each of the given numbers is a multiple of 3:3:

  1. 645 645
  2. 10,51910,519
Try It 2.85

Determine whether each number is a multiple of 3:3:

  1. 954 954
  2. 3,7423,742
Try It 2.86

Determine whether each number is a multiple of 3:3:

  1. 643 643
  2. 8,3798,379

Look back at the charts where you highlighted the multiples of 2,2, of 5,5, and of 10.10. Notice that the multiples of 1010 are the numbers that are multiples of both 22 and 5.5. That is because 10=25.10=25. Likewise, since 6=23,6=23, the multiples of 66 are the numbers that are multiples of both 22 and 3.3.

Use Common Divisibility Tests

Another way to say that 375375 is a multiple of 55 is to say that 375375 is divisible by 5.5. In fact, 375÷5375÷5 is 75,75, so 375375 is 575.575. Notice in Example 2.43 that 10,51910,519 is not a multiple 3.3. When we divided 10,51910,519 by 33 we did not get a counting number, so 10,51910,519 is not divisible by 3.3.

Divisibility

If a number mm is a multiple of n,n, then we say that mm is divisible by n.n.

Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests. Table 2.10 summarizes divisibility tests for some of the counting numbers between one and ten.

Divisibility Tests
A number is divisible by
22 if the last digit is 0,2,4,6,or80,2,4,6,or8
33 if the sum of the digits is divisible by 33
55 if the last digit is 55 or 00
66 if divisible by both 22 and 33
1010 if the last digit is 00
Table 2.10

Example 2.44

Determine whether 1,2901,290 is divisible by 2,3,5,and10.2,3,5,and10.

Try It 2.87

Determine whether the given number is divisible by 2,3,5,and10.2,3,5,and10.

62406240

Try It 2.88

Determine whether the given number is divisible by 2,3,5,and10.2,3,5,and10.

72487248

Example 2.45

Determine whether 5,6255,625 is divisible by 2,3,5,and10.2,3,5,and10.

Try It 2.89

Determine whether the given number is divisible by2,3,5,and10.by2,3,5,and10.

49624962

Try It 2.90

Determine whether the given number is divisible by2,3,5,and10.by2,3,5,and10.

37653765

Find All the Factors of a Number

There are often several ways to talk about the same idea. So far, we’ve seen that if mm is a multiple of n,n, we can say that mm is divisible by n.n. We know that 7272 is the product of 88 and 9,9, so we can say 7272 is a multiple of 88 and 7272 is a multiple of 9.9. We can also say 7272 is divisible by 88 and by 9.9. Another way to talk about this is to say that 88 and 99 are factors of 72.72. When we write 72=8972=89 we can say that we have factored 72.72.

The image shows the equation 8 times 9 equals 72. The 8 and 9 are labeled as factors and the 72 is labeled product.

Factors

If ab=m,ab=m, then aandbaandb are factors of m,m, and mm is the product of aandb.aandb.

In algebra, it can be useful to determine all of the factors of a number. This is called factoring a number, and it can help us solve many kinds of problems.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Model Multiplication and Factoring” will help you develop a better understanding of multiplication and factoring.

For example, suppose a choreographer is planning a dance for a ballet recital. There are 2424 dancers, and for a certain scene, the choreographer wants to arrange the dancers in groups of equal sizes on stage.

In how many ways can the dancers be put into groups of equal size? Answering this question is the same as identifying the factors of 24.24. Table 2.13 summarizes the different ways that the choreographer can arrange the dancers.

Number of Groups Dancers per Group Total Dancers
11 2424 124=24124=24
22 1212 212=24212=24
33 88 38=2438=24
44 66 46=2446=24
66 44 64=2464=24
88 33 83=2483=24
1212 22 122=24122=24
2424 11 241=24241=24
Table 2.13

What patterns do you see in Table 2.13? Did you notice that the number of groups times the number of dancers per group is always 24?24? This makes sense, since there are always 2424 dancers.

You may notice another pattern if you look carefully at the first two columns. These two columns contain the exact same set of numbers—but in reverse order. They are mirrors of one another, and in fact, both columns list all of the factors of 24,24, which are:

1,2,3,4,6,8,12,241,2,3,4,6,8,12,24

We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with 1.1. If the quotient is also a counting number, then the divisor and the quotient are factors of the number. We can stop when the quotient becomes smaller than the divisor.

How To

Find all the factors of a counting number.

  1. Step 1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
    • If the quotient is a counting number, the divisor and quotient are a pair of factors.
    • If the quotient is not a counting number, the divisor is not a factor.
  2. Step 2. List all the factor pairs.
  3. Step 3. Write all the factors in order from smallest to largest.

Example 2.46

Find all the factors of 72.72.

Try It 2.91

Find all the factors of the given number:

9696

Try It 2.92

Find all the factors of the given number:

8080

Identify Prime and Composite Numbers

Some numbers, like 72,72, have many factors. Other numbers, such as 7,7, have only two factors: 11 and the number. A number with only two factors is called a prime number. A number with more than two factors is called a composite number. The number 11 is neither prime nor composite. It has only one factor, itself.

Prime Numbers and Composite Numbers

A prime number is a counting number greater than 11 whose only factors are 11 and itself.

A composite number is a counting number that is not prime.

Figure 2.10 lists the counting numbers from 22 through 2020 along with their factors. The highlighted numbers are prime, since each has only two factors.

This figure shows a table with twenty rows and three columns. The first row is a header row. It labels the columns as “Number”, “Factor” and “Prime or composite?” The second row lists the number 2, in red, under the “Number” column, the numbers 1 and 2 under the “Factors” column and the word prime under the “Prime or Composite?” column. The third row lists the number 3, in red, under the “Number” column, the numbers 1 and 3 under the “Factors” column and the word prime under the “Prime or Composite?” column. The fourth row lists the number 4 under the “Number” column, the numbers 1, 2 and 4 under the “Factors” column and the word composite under the “Prime or Composite?” column. The fifth row lists the number 5, in red, under the “Number” column, the numbers 1 and 5 under the “Factors” column and the word prime under the “Prime or Composite?” column. The sixth row lists the number 6 under the “Number” column, the numbers 1, 2, 3 and 6 under the “Factors” column and the word composite under the “Prime or Composite?” column. The seventh row lists the number 7, in red, under the “Number” column, the numbers 1 and 7 under the “Factors” column and the word prime under the “Prime or Composite?” column. The eighth row lists the number 8 under the “Number” column, the numbers 1, 2, 4 and 8 under the “Factors” column and the word composite under the “Prime or Composite?” column. The ninth row lists the number 9 under the “Number” column, the numbers 1, 3 and 9 under the “Factors” column and the word composite under the “Prime or Composite?” column. The tenth row lists the number 10 under the “Number” column, the numbers 1, 2, 5 and 10 under the “Factors” column and the word composite under the “Prime or Composite?” column. The eleventh row lists the number 11, in red, under the “Number” column, the numbers 1 and 11 under the “Factors” column and the word prime under the “Prime or Composite?” column. The twelfth row lists the number 12 under the “Number” column, the numbers 1, 2, 3, 4, 6 and 12 under the “Factors” column and the word composite under the “Prime or Composite?” column. The thirteenth row lists the number 13, in red, under the “Number” column, the numbers 1 and 13 under the “Factors” column and the word prime under the “Prime or Composite?” column. The fourteenth row lists the number 14 under the “Number” column, the numbers 1, 2, 7 and 14 under the “Factors” column and the word composite under the “Prime or Composite?” column. The fifteenth row lists the number 15 under the “Number” column, the numbers 1, 2, 3, 5 and 15 under the “Factors” column and the word composite under the “Prime or Composite?” column. The sixteenth row lists the number 16 under the “Number” column, the numbers 1, 2, 4, 8 and 16 under the “Factors” column and the word composite under the “Prime or Composite?” column. The seventeenth row lists the number 17, in red, under the “Number” column, the numbers 1 and 17 under the “Factors” column and the word prime under the “Prime or Composite?” column. The eighteenth row lists the number 18 under the “Number” column, the numbers 1, 2, 3, 6, 9 and 18 under the “Factors” column and the word composite under the “Prime or Composite?” column. The nineteenth row lists the number 19, in red, under the “Number” column, the numbers 1 and 19 under the “Factors” column and the word prime under the “Prime or Composite?” column. The twentieth row lists the number 20 under the “Number” column, the numbers 1, 2, 4, 5, 10 and 20 under the “Factors” column and the word composite under the “Prime or Composite?” column.
Figure 2.10 Factors of the counting numbers from 22 through 20,20, with prime numbers highlighted

The prime numbers less than 2020 are 2,3,5,7,11,13,17,and19.2,3,5,7,11,13,17,and19. There are many larger prime numbers too. In order to determine whether a number is prime or composite, we need to see if the number has any factors other than 11 and itself. To do this, we can test each of the smaller prime numbers in order to see if it is a factor of the number. If none of the prime numbers are factors, then that number is also prime.

How To

Determine if a number is prime.

  1. Step 1. Test each of the primes, in order, to see if it is a factor of the number.
  2. Step 2. Start with 22 and stop when the quotient is smaller than the divisor or when a prime factor is found.
  3. Step 3. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.

Example 2.47

Identify each number as prime or composite:

  1. 8383
  2. 7777
Try It 2.93

Identify the number as prime or composite:

9191

Try It 2.94

Identify the number as prime or composite:

137137

Section 2.4 Exercises

Practice Makes Perfect

Identify Multiples of Numbers

In the following exercises, list all the multiples less than 5050 for the given number.

215.

22

216.

33

217.

44

218.

55

219.

66

220.

77

221.

88

222.

99

223.

1010

224.

1212

Use Common Divisibility Tests

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2,3,4,5,6,and10.2,3,4,5,6,and10.

225.

8484

226.

9696

227.

7575

228.

7878

229.

168168

230.

264264

231.

900900

232.

800800

233.

896896

234.

942942

235.

375375

236.

750750

237.

350350

238.

550550

239.

14301430

240.

10801080

241.

22,33522,335

242.

39,07539,075

Find All the Factors of a Number

In the following exercises, find all the factors of the given number.

243.

3636

244.

4242

245.

6060

246.

4848

247.

144144

248.

200200

249.

588588

250.

576576

Identify Prime and Composite Numbers

In the following exercises, determine if the given number is prime or composite.

251.

4343

252.

6767

253.

3939

254.

5353

255.

7171

256.

119119

257.

481481

258.

221221

259.

209209

260.

359359

261.

667667

262.

17711771

Everyday Math

263.

Banking Frank’s grandmother gave him $100$100 at his high school graduation. Instead of spending it, Frank opened a bank account. Every week, he added $15$15 to the account. The table shows how much money Frank had put in the account by the end of each week. Complete the table by filling in the blanks.

Weeks after graduation Total number of dollars Frank put in the account Simplified Total
00 100100 100100
11 100+15100+15 115115
22 100+152100+152 130130
33 100+153100+153
44 100+15[]100+15[]
55 100+[]100+[]
66
2020
xx
264.

Banking In March, Gina opened a Christmas club savings account at her bank. She deposited $75$75 to open the account. Every week, she added $20$20 to the account. The table shows how much money Gina had put in the account by the end of each week. Complete the table by filling in the blanks.

Weeks after opening the account Total number of dollars Gina put in the account Simplified Total
00 7575 7575
11 75+2075+20 9595
22 75+20275+202 115115
33 75+20375+203
44 75+20[]75+20[]
55 75+[]75+[]
66
2020
xx

Writing Exercises

265.

If a number is divisible by 22 and by 3,3, why is it also divisible by 6?6?

266.

What is the difference between prime numbers and composite numbers?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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