2.5: Find Multiples and Factors
- Page ID
- 114871
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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)?
If you missed this problem, review Example 1.1.
Be Prepared 2.11
Find the sum of and
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: She has shoes in her closet.
The numbers are called multiples of Multiples of can be written as the product of a counting number and The first six multiples of are given below.
A multiple of a number is the product of the number and a counting number. So a multiple of would be the product of a counting number and Below are the first six multiples of
We can find the multiples of any number by continuing this process. Table 2.8 shows the multiples of through for the first twelve counting numbers.
Counting Number | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Multiple of a Number
A number is a multiple of if it is the product of a counting number and
Recognizing the patterns for multiples of 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 to Multiples of are highlighted. Do you notice a pattern?
The last digit of each highlighted number in Figure 2.6 is either This is true for the product of and any counting number. So, to tell if any number is a multiple of look at the last digit. If it is then the number is a multiple of
Example 2.40
Determine whether each of the following is a multiple of
- ⓐ
- ⓑ
- Answer
ⓐ Is 489 a multiple of 2? Is the last digit 0, 2, 4, 6, or 8? No. 489 is not a multiple of 2. ⓑ Is 3,714 a multiple of 2? Is the last digit 0, 2, 4, 6, or 8? Yes. 3,714 is a multiple of 2.
Try It 2.79
Determine whether each number is a multiple of
- ⓐ
- ⓑ
Try It 2.80
Determine whether each number is a multiple of
- ⓐ
- ⓑ
Now let’s look at multiples of
All multiples of
Example 2.41
Determine whether each of the following is a multiple of
- ⓐ
579 579 - ⓑ
880 880
- Answer
ⓐ Is 579 a multiple of 5? Is the last digit 5 or 0? No. 579 is not a multiple of 5. ⓑ Is 880 a multiple of 5? Is the last digit 5 or 0? Yes. 880 is a multiple of 5.
Try It 2.81
Determine whether each number is a multiple of
- ⓐ
675 675 - ⓑ
1,578 1,578
Try It 2.82
Determine whether each number is a multiple of
- ⓐ
421 421 - ⓑ
2,690 2,690
Figure 2.8 highlights the multiples of
Example 2.42
Determine whether each of the following is a multiple of
- ⓐ
425 425 - ⓑ
350 350
- Answer
ⓐ Is 425 a multiple of 10? Is the last digit zero? No. 425 is not a multiple of 10. ⓑ Is 350 a multiple of 10? Is the last digit zero? Yes. 350 is a multiple of 10.
Try It 2.83
Determine whether each number is a multiple of
- ⓐ
179 179 - ⓑ
3,540 3,540
Try It 2.84
Determine whether each number is a multiple of
- ⓐ
110 110 - ⓑ
7,595 7,595
Figure 2.9 highlights multiples of
Unlike the other patterns we’ve examined so far, this pattern does not involve the last digit. The pattern for multiples of
Consider the number
Example 2.43
Determine whether each of the given numbers is a multiple of
- ⓐ
645 645 - ⓑ
10,519 10,519
- Answer
ⓐ Is
a multiple of645 645 3 ? 3 ? Find the sum of the digits. 6 + 4 + 5 = 15 6 + 4 + 5 = 15 Is 15 a multiple of 3? Yes. If we're not sure, we could add its digits to find out. We can check it by dividing 645 by 3. 645 ÷ 3 645 ÷ 3 The quotient is 215. 3 ⋅ 215 = 645 3 ⋅ 215 = 645 ⓑ Is
a multiple of10,519 10,519 3 ? 3 ? Find the sum of the digits. 1 + 0 + 5 + 1 + 9 = 16 1 + 0 + 5 + 1 + 9 = 16 Is 16 a multiple of 3? No. So 10,519 is not a multiple of 3 either.. 645 ÷ 3 645 ÷ 3 We can check this by dividing by 10,519 by 3. 3,506 R 1 3 10,519 3,506 R 1 3 10,519
When we divide
Try It 2.85
Determine whether each number is a multiple of
- ⓐ
954 954 - ⓑ
3,742 3,742
Try It 2.86
Determine whether each number is a multiple of
- ⓐ
643 643 - ⓑ
8,379 8,379
Look back at the charts where you highlighted the multiples of
Use Common Divisibility Tests
Another way to say that
Divisibility
If a number
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 | |
if the last digit is |
|
if the sum of the digits is divisible by |
|
if the last digit is |
|
if divisible by both |
|
if the last digit is |
Example 2.44
Determine whether
- Answer
Table 2.11 applies the divisibility tests to
In the far right column, we check the results of the divisibility tests by seeing if the quotient is a whole number.1,290 . 1,290 . Divisible by…? Test Divisible? Check 2 2 Is last digit Yes.0 , 2 , 4 , 6 , or 8 ? 0 , 2 , 4 , 6 , or 8 ? yes 1290 ÷ 2 = 645 1290 ÷ 2 = 645 3 3 Is sum of digits divisible by 3 ? Is sum of digits divisible by 3 ?
Yes.1 + 2 + 9 + 0 = 12 1 + 2 + 9 + 0 = 12 yes 1290 ÷ 3 = 430 1290 ÷ 3 = 430 5 5 Is last digit or5 5 Yes.0 ? 0 ? yes 1290 ÷ 5 = 258 1290 ÷ 5 = 258 10 10 Is last digit Yes.0 ? 0 ? yes 1290 ÷ 10 = 129 1290 ÷ 10 = 129
Thus,
Try It 2.87
Determine whether the given number is divisible by
Try It 2.88
Determine whether the given number is divisible by
Example 2.45
Determine whether
- Answer
Table 2.12 applies the divisibility tests to
and tests the results by finding the quotients.5,625 5,625 Divisible by…? Test Divisible? Check 2 2 Is last digit No.0 , 2 , 4 , 6 , or 8 ? 0 , 2 , 4 , 6 , or 8 ? no 5625 ÷ 2 = 2812.5 5625 ÷ 2 = 2812.5 3 3 Is sum of digits divisible by 3 ? Is sum of digits divisible by 3 ?
Yes.5 + 6 + 2 + 5 = 18 5 + 6 + 2 + 5 = 18 yes 5625 ÷ 3 = 1875 5625 ÷ 3 = 1875 5 5 Is last digit is or5 5 Yes.0 ? 0 ? yes 5625 ÷ 5 = 1125 5625 ÷ 5 = 1125 10 10 Is last digit No.0 ? 0 ? no 5625 ÷ 10 = 562.5 5625 ÷ 10 = 562.5
Thus,
Try It 2.89
Determine whether the given number is divisible
Try It 2.90
Determine whether the given number is divisible
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
Factors
In the expression
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
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
Number of Groups | Dancers per Group | Total Dancers |
---|---|---|
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
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
We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with
How To
Find all the factors of a counting number.
- 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.
- Step 2. List all the factor pairs.
- Step 3. Write all the factors in order from smallest to largest.
Example 2.46
Find all the factors of
- Answer
Divide
by each of the counting numbers starting with72 72 If the quotient is a whole number, the divisor and quotient are a pair of factors.1 . 1 . The next line would have a divisor of
and a quotient of9 9 The quotient would be smaller than the divisor, so we stop. If we continued, we would end up only listing the same factors again in reverse order. Listing all the factors from smallest to greatest, we have8 . 8 . 1 , 2 , 3 , 4 , 6 , 8 , 9 , 12 , 18 , 24 , 36 , and 72 1 , 2 , 3 , 4 , 6 , 8 , 9 , 12 , 18 , 24 , 36 , and 72
Try It 2.91
Find all the factors of the given number:
Try It 2.92
Find all the factors of the given number:
Identify Prime and Composite Numbers
Some numbers, like
Prime Numbers and Composite Numbers
A prime number is a counting number greater than
A composite number is a counting number that is not prime.
Figure 2.10 lists the counting numbers from
The prime numbers less than
How To
Determine if a number is prime.
- Step 1. Test each of the primes, in order, to see if it is a factor of the number.
- Step 2. Start with
and stop when the quotient is smaller than the divisor or when a prime factor is found.2 2 - 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:
- ⓐ
83 83 - ⓑ
77 77
- Answer
ⓐ Test each prime, in order, to see if it is a factor of
, starting with83 83 as shown. We will stop when the quotient is smaller than the divisor.2 , 2 , Prime Test Factor of 83 ? 83 ? 2 2 Last digit of is not83 83 0 , 2 , 4 , 6 , or 8 . 0 , 2 , 4 , 6 , or 8 . No. 3 3 and8 + 3 = 11 , 8 + 3 = 11 , is not divisible by11 11 3 . 3 . No. 5 5 The last digit of is not83 83 or5 5 0 . 0 . No. 7 7 83 ÷ 7 = 11.857 …. 83 ÷ 7 = 11.857 …. No. 11 11 83 ÷ 11 = 7.545 … 83 ÷ 11 = 7.545 … No. We can stop when we get to
because the quotient11 11 is less than the divisor.(7.545…) (7.545…) We did not find any prime numbers that are factors of
so we know83 , 83 , is prime.83 83 ⓑ Test each prime, in order, to see if it is a factor of
77 . 77 . Prime Test Factor of 77 ? 77 ? 2 2 Last digit is not 0 , 2 , 4 , 6 , or 8 . 0 , 2 , 4 , 6 , or 8 . No. 3 3 and7 + 7 = 14 , 7 + 7 = 14 , is not divisible by14 14 3 . 3 . No. 5 5 the last digit is not or5 5 0 . 0 . No. 7 7 77 ÷ 7 = 11 77 ÷ 7 = 11 Yes.
Since
Try It 2.93
Identify the number as prime or composite:
Try It 2.94
Identify the number as prime or composite:
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Section 2.4 Exercises
Practice Makes Perfect
Identify Multiples of Numbers
In the following exercises, list all the multiples less than
Use Common Divisibility Tests
In the following exercises, use the divisibility tests to determine whether each number is divisible by
Find All the Factors of a Number
In the following exercises, find all the factors of the given number.
Identify Prime and Composite Numbers
In the following exercises, determine if the given number is prime or composite.
Everyday Math
Banking Frank’s grandmother gave him
Weeks after graduation | Total number of dollars Frank put in the account | Simplified Total |
---|---|---|
Banking In March, Gina opened a Christmas club savings account at her bank. She deposited
Weeks after opening the account | Total number of dollars Gina put in the account | Simplified Total |
---|---|---|
Writing Exercises
If a number is divisible by
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?