2.4 Find Multiples and Factors

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: $2, 4, 6, 8, 10, 12.$ She has $12$ shoes in her closet.

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

$\begin{array}{l} 1 \cdot 2 = 2 \\ 2 \cdot 2 = 4 \\ 3 \cdot 2 = 6 \\ 4 \cdot 2 = 8 \\ 5 \cdot 2 = 10 \\ 6 \cdot 2 = 12 \end{array}$

A multiple of a number is the product of the number and a counting number. So a multiple of $3$ would be the product of a counting number and $3 .$ Below are the first six multiples of $3 .$

$\begin{array}{l} 1 \cdot 3 = 3 \\ 2 \cdot 3 = 6 \\ 3 \cdot 3 = 9 \\ 4 \cdot 3 = 12 \\ 5 \cdot 3 = 15 \\ 6 \cdot 3 = 18 \end{array}$

We can find the multiples of any number by continuing this process. Table 2.9 shows the multiples of $2$ through $9$ for the first twelve counting numbers.

**Table 2.9**
Counting Number	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$
$Multiples of 2$	$2$	$4$	$6$	$8$	$10$	$12$	$14$	$16$	$18$	$20$	$22$	$24$
$Multiples of 3$	$3$	$6$	$9$	$12$	$15$	$18$	$21$	$24$	$27$	$30$	$33$	$36$
$Multiples of 4$	$4$	$8$	$12$	$16$	$20$	$24$	$28$	$32$	$36$	$40$	$44$	$48$
$Multiples of 5$	$5$	$10$	$15$	$20$	$25$	$30$	$35$	$40$	$45$	$50$	$55$	$60$
$Multiples of 6$	$6$	$12$	$18$	$24$	$30$	$36$	$42$	$48$	$54$	$60$	$66$	$72$
$Multiples of 7$	$7$	$14$	$21$	$28$	$35$	$42$	$49$	$56$	$63$	$70$	$77$	$84$
$Multiples of 8$	$8$	$16$	$24$	$32$	$40$	$48$	$56$	$64$	$72$	$80$	$88$	$96$
$Multiples of 9$	$9$	$18$	$27$	$36$	$45$	$54$	$63$	$72$	$81$	$90$	$99$	$108$

Recognizing the patterns for multiples of $2, 5, 10, and 3$ will be helpful to you as you continue in this course.

Figure 2.9 shows the counting numbers from $1$ to $50 .$ Multiples of $2$ 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 2.9** Multiples of $2$ between $1$ and $50$

The last digit of each highlighted number in Figure 2.9 is either $0, 2, 4, 6, or 8.$ This is true for the product of $2$ and any counting number. So, to tell if any number is a multiple of $2$ look at the last digit. If it is $0, 2, 4, 6, or 8,$ then the number is a multiple of $2 .$

Example 1

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

ⓐ $489$
ⓑ $3,714$

ⓐ
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.

Now let’s look at multiples of $5 .$ Figure 2.10 highlights all of the multiples of $5$ between $1$ and $50 .$ What do you notice about the multiples of $5 ?$

All multiples of $5$ end with either $5$ or $0 .$ Just like we identify multiples of $2$ by looking at the last digit, we can identify multiples of $5$ by looking at the last digit.

Example 2

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

ⓐ $579$
ⓑ $880$

ⓐ
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.

Figure 2.11 highlights the multiples of $10$ between $1$ and $50 .$ All multiples of $10$ all end with a zero.

Example 3

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

ⓐ $425$
ⓑ $350$

ⓐ
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.

Figure 2.12 highlights multiples of $3 .$ The pattern for multiples of $3$ is not as obvious as the patterns for multiples of $2, 5, and 10.$

Unlike the other patterns we’ve examined so far, this pattern does not involve the last digit. The pattern for multiples of $3$ is based on the sum of the digits. If the sum of the digits of a number is a multiple of $3,$ then the number itself is a multiple of $3 .$ See Table 2.10.

**Table 2.10**
$Multiple of 3$	$3$	$6$	$9$	$12$	$15$	$18$	$21$	$24$
$Sum of digits$	$3$	$6$	$9$	$1 plus 2; 3$	$1 plus 5; 6$	$1 plus 8; 9$	$2 plus 1; 3$	$2 plus 4; 6$

Consider the number $42 .$ The digits are $4$ and $2,$ and their sum is $4 + 2 = 6 .$ Since $6$ is a multiple of $3,$ we know that $42$ is also a multiple of $3 .$

Example 4

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

ⓐ $645$
ⓑ $10,519$

ⓐ Is $645$ a multiple of $3 ?$

Find the sum of the digits.	$6 + 4 + 5 = 15$
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.	$645 divided by 3$
The quotient is 215.	$3 \cdot 215 = 645$

ⓑ Is $10,519$ a multiple of $3 ?$

Find the sum of the digits.	$1 + 0 + 5 + 1 + 9 = 16$
Is 16 a multiple of 3?	No.
So 10,519 is not a multiple of 3 either..	$645 divided by 3$
We can check this by dividing by 10,519 by 3.	$3,506 R 1; 3 10,519$

When we divide $10,519$ by $3,$ we do not get a counting number, so $10,519$ is not the product of a counting number and $3 .$ It is not a multiple of $3 .$

Look back at the charts where you highlighted the multiples of $2,$ of $5,$ and of $10 .$ Notice that the multiples of $10$ are the numbers that are multiples of both $2$ and $5 .$ That is because $10 = 2 \cdot 5 .$ Likewise, since $6 = 2 \cdot 3,$ the multiples of $6$ are the numbers that are multiples of both $2$ and $3 .$

Use Common Divisibility Tests

Another way to say that $375$ is a multiple of $5$ is to say that $375$ is divisible by $5 .$ In fact, $375 divided by 5$ is $75,$ so $375$ is $5 times 75.$ Notice in Example 4 that $10,519$ is not a multiple $3 .$ When we divided $10,519$ by $3$ we did not get a counting number, so $10,519$ is not divisible by $3 .$

Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests. Table 2.11 summarizes divisibility tests for some of the counting numbers between one and ten.

**Table 2.11**
Divisibility Tests
A number is divisible by
$2$	if the last digit is $0, 2, 4, 6, or 8$
$3$	if the sum of the digits is divisible by $3$
$5$	if the last digit is $5$ or $0$
$6$	if divisible by both $2$ and $3$
$10$	if the last digit is $0$

Example 5

Determine whether $1,290$ is divisible by $2, 3, 5, and 10.$

Table 2.12 applies the divisibility tests to $1,290.$ In the far right column, we check the results of the divisibility tests by seeing if the quotient is a whole number.

**Table 2.12**
Divisible by…?	Test	Divisible?	Check
$2$	Is last digit $0, 2, 4, 6, or 8 ?$ Yes.	yes	$1290 \div 2 = 645$
$3$	$Is sum of digits divisible by 3 ?$ $1 + 2 + 9 + 0 = 12$ Yes.	yes	$1290 \div 3 = 430$
$5$	Is last digit $5$ or $0 ?$ Yes.	yes	$1290 \div 5 = 258$
$10$	Is last digit $0 ?$ Yes.	yes	$1290 \div 10 = 129$

Thus, $1,290$ is divisible by $2, 3, 5, and 10.$

Example 6

Determine whether $5,625$ is divisible by $2, 3, 5, and 10.$

Table 2.13 applies the divisibility tests to $5,625$ and tests the results by finding the quotients.

**Table 2.13**
Divisible by…?	Test	Divisible?	Check
$2$	Is last digit $0, 2, 4, 6, or 8 ?$ No.	no	$5625 \div 2 = 2812.5$
$3$	$Is sum of digits divisible by 3 ?$ $5 + 6 + 2 + 5 = 18$ Yes.	yes	$5625 \div 3 = 1875$
$5$	Is last digit is $5$ or $0 ?$ Yes.	yes	$5625 \div 5 = 1125$
$10$	Is last digit $0 ?$ No.	no	$5625 \div 10 = 562.5$

Thus, $5,625$ is divisible by $3$ and $5,$ but not $2,$ or $10 .$

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 $m$ is a multiple of $n,$ we can say that $m$ is divisible by $n .$ We know that $72$ is the product of $8$ and $9,$ so we can say $72$ is a multiple of $8$ and $72$ is a multiple of $9 .$ We can also say $72$ is divisible by $8$ and by $9 .$ Another way to talk about this is to say that $8$ and $9$ are factors of $72 .$ When we write $72 = 8 \cdot 9$ we can say that we have factored $72 .$

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

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.

For example, suppose a choreographer is planning a dance for a ballet recital. There are $24$ 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 $24 .$ Table 2.14 summarizes the different ways that the choreographer can arrange the dancers.

**Table 2.14**
Number of Groups	Dancers per Group	Total Dancers
$1$	$24$	$1 \cdot 24 = 24$
$2$	$12$	$2 \cdot 12 = 24$
$3$	$8$	$3 \cdot 8 = 24$
$4$	$6$	$4 \cdot 6 = 24$
$6$	$4$	$6 \cdot 4 = 24$
$8$	$3$	$8 \cdot 3 = 24$
$12$	$2$	$12 \cdot 2 = 24$
$24$	$1$	$24 \cdot 1 = 24$

What patterns do you see in Table 2.14? Did you notice that the number of groups times the number of dancers per group is always $24 ?$ This makes sense, since there are always $24$ 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 $24,$ which are:

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

We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with $1 .$ 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.

Identify Prime and Composite Numbers

Some numbers, like $72,$ have many factors. Other numbers, such as $7,$ have only two factors: $1$ 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 $1$ is neither prime nor composite. It has only one factor, itself.

Figure 2.13 lists the counting numbers from $2$ through $20$ 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 2.13** Factors of the counting numbers from $2$ through $20,$ with prime numbers highlighted

The prime numbers less than $20$ are $2, 3, 5, 7, 11, 13, 17, and 19.$ 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 $1$ 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.

Example 8

Identify each number as prime or composite:

ⓐ $83$
ⓑ $77$

ⓐ Test each prime, in order, to see if it is a factor of $83$ , starting with $2,$ as shown. We will stop when the quotient is smaller than the divisor.

Prime	Test	Factor of $83 ?$
$2$	Last digit of $83$ is not $0, 2, 4, 6, or 8.$	No.
$3$	$8 + 3 = 11,$ and $11$ is not divisible by $3 .$	No.
$5$	The last digit of $83$ is not $5$ or $0 .$	No.
$7$	$83 \div 7 = 11.857 ….$	No.
$11$	$83 \div 11 = 7.545 …$	No.

We can stop when we get to $11$ because the quotient $(7.545…)$ is less than the divisor.

We did not find any prime numbers that are factors of $83,$ so we know $83$ is prime.

ⓑ Test each prime, in order, to see if it is a factor of $77 .$

Prime	Test	Factor of $77 ?$
$2$	Last digit is not $0, 2, 4, 6, or 8.$	No.
$3$	$7 + 7 = 14,$ and $14$ is not divisible by $3 .$	No.
$5$	the last digit is not $5$ or $0 .$	No.
$7$	$77 \div 7 = 11$	Yes.

Since $77$ is divisible by $7,$ we know it is not a prime number. It is composite.

Key Concepts

Divisibility Tests
A number is divisible by
2	if the last digit is 0, 2, 4, 6, or 8
3	if the sum of the digits is divisible by 3
4	if the last two digits are a number divisible by 4
5	if the last digit is 5 or 0
6	if divisible by both 2 and 3
10	if the last digit is 0

Factors If $a \cdot b = m$ , then $a$ and $b$ are factors of $m$ , and $m$ is the product of $a$ and $b$ .
Find all the factors of a counting number.
1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
  1. If the quotient is a counting number, the divisor and quotient are a pair of factors.
  2. If the quotient is not a counting number, the divisor is not a factor.
2. List all the factor pairs.
3. Write all the factors in order from smallest to largest.
Determine if a number is prime.
1. Test each of the primes, in order, to see if it is a factor of the number.
2. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found.
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.

Practice Makes Perfect

Identify Multiples of Numbers

In the following exercises, list all the multiples less than $50$ for the given number.

$2$

2, 4, 6, 8, 10 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48

$3$

$4$

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

$5$

$6$

6, 12, 18, 24, 30, 36, 42, 48

$7$

$8$

8, 16, 24, 32, 40, 48

$9$

$10$

10, 20, 30, 40

$12$

Use Common Divisibility Tests

In the following exercises, use the divisibility tests to determine whether each number is divisible by $2, 3, 4, 5, 6, and 10.$

$84$

Divisible by 2, 3, 4, 6

$96$

$75$

Divisible by 3, 5

$78$

$168$

Divisible by 2, 3, 4, 6

$264$

$900$

Divisible by 2, 3, 4, 5, 6, 10

$800$

$896$

Divisible by 2, 4

$942$

$375$

Divisible by 3, 5

$750$

$350$

Divisible by 2, 5, 10

$550$

$1430$

Divisible by 2, 5, 10

$1080$

$22,335$

Divisible by 3, 5

$39,075$

Find All the Factors of a Number

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

$36$

1, 2, 3, 4, 6, 9, 12, 18, 36

$42$

$60$

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

$48$

$144$

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72,144

$200$

$588$

1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588

$576$

Identify Prime and Composite Numbers

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

$43$

prime

$67$

$39$

composite

$53$

$71$

prime

$119$

$481$

composite

$221$

$209$

composite

$359$

$667$

composite

$1771$

Everyday Math

Banking Frank’s grandmother gave him $$100$ at his high school graduation. Instead of spending it, Frank opened a bank account. Every week, he added $$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
$0$	$100$	$100$
$1$	$100 plus 15$	$115$
$2$	$100 + 15 \cdot 2$	$130$
$3$	$100 + 15 \cdot 3$
$4$	$100 + 15 \cdot []$
$5$	$100 plus$
$6$
$20$
$x$

This table has nine rows and three columns. The first row is a header row that labels each column. The first column is labeled “Weeks after opening the account”, the second is labeled “Total number of dollars Gina put in the account”, and the last is labeled “Simplified Total”. Under the “Weeks after opening the account” column are the values: 0, 1, 2, 3, 4, 5, 6, 20, and the letter x. Under the “Total number of dollars Gina put in the account” column are the expressions: 75; 75 plus 20; 75 plus 20 times 2; 75 plus 20 times 3; 75 plus 20 times empty set of brackets; 75 plus empty set of brackets; the last three rows are blank. Under the “Simplified Total” column are the values: 75, 95, 115, the last six rows are blank.

Banking In March, Gina opened a Christmas club savings account at her bank. She deposited $$75$ to open the account. Every week, she added $$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
$0$	$75$	$75$
$1$	$75 plus 20$	$95$
$2$	$75 + 20 \cdot 2$	$115$
$3$	$75 + 20 \cdot 3$
$4$	$75 + 20 \cdot []$
$5$	$75 plus$
$6$
$20$
$x$

Writing Exercises

If a number is divisible by $2$ and by $3,$ why is it also divisible by $6 ?$

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.

Self-assessment grid for math skills in number theory: identifying multiples, using divisibility tests, finding factors, and identifying prime and composite numbers. Students rate their understanding. — **Figure 2.14**

ⓑ 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?