Prealgebra 2e

# 2.4Find Multiples and Factors

Prealgebra 2e2.4 Find Multiples and Factors

### 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.

$1⋅2=22⋅2=43⋅2=64⋅2=85⋅2=106⋅2=121⋅2=22⋅2=43⋅2=64⋅2=85⋅2=106⋅2=12$

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 $3.3.$

$1⋅3=32⋅3=63⋅3=94⋅3=125⋅3=156⋅3=181⋅3=32⋅3=63⋅3=94⋅3=125⋅3=156⋅3=18$

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?

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 $5?5?$

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.

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 $2,5,and10.2,5,and10.$

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 $10=2⋅5.10=2⋅5.$ Likewise, since $6=2⋅3,6=2⋅3,$ 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 $5⋅75.5⋅75.$ 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 $72=8⋅972=8⋅9$ we can say that we have factored $72.72.$

### Factors

In the expression $a⋅ba⋅b$, both a and b are called factors. If $a⋅b=m,a⋅b=m,$ and both a and b are integers, 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
$11$ $2424$ $1⋅24=241⋅24=24$
$22$ $1212$ $2⋅12=242⋅12=24$
$33$ $88$ $3⋅8=243⋅8=24$
$44$ $66$ $4⋅6=244⋅6=24$
$66$ $44$ $6⋅4=246⋅4=24$
$88$ $33$ $8⋅3=248⋅3=24$
$1212$ $22$ $12⋅2=2412⋅2=24$
$2424$ $11$ $24⋅1=2424⋅1=24$
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.

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 $100100$ at his high school graduation. Instead of spending it, Frank opened a bank account. Every week, he added $1515$ 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
$00$ $100100$ $100100$
$11$ $100+15100+15$ $115115$
$22$ $100+15⋅2100+15⋅2$ $130130$
$33$ $100+15⋅3100+15⋅3$
$44$ $100+15⋅[]100+15⋅[]$
$55$ $100+[]100+[]$
$66$
$2020$
$xx$
264.

Banking In March, Gina opened a Christmas club savings account at her bank. She deposited $7575$ to open the account. Every week, she added $2020$ 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
$00$ $7575$ $7575$
$11$ $75+2075+20$ $9595$
$22$ $75+20⋅275+20⋅2$ $115115$
$33$ $75+20⋅375+20⋅3$
$44$ $75+20⋅[]75+20⋅[]$
$55$ $75+[]75+[]$
$66$
$2020$
$xx$

#### 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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