Identifying Multiples of Numbers

Learning Outcomes

Determine whether a given number is divisible by 2, 3, 5, or 10

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. The table below shows the multiples of $2$ through $9$ for the first twelve counting numbers.

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

Multiple of a Number

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

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

Doing the Manipulative Mathematics activity “Multiples” will help you develop a better understanding of multiples.
The table below shows the counting numbers from $1$ to $50$ . Multiples of $2$ are highlighted. Do you notice a pattern?

Multiples of $2$ between $1$ and $50$

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.

The last digit of each highlighted number in the table is either $0, 2, 4, 6, \text{or }8$ . 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 $0, 2, 4, 6, \text{or }8$ , then the number is a multiple of $2$ .

example

Determine whether each of the following is a multiple of $2\text{:}$

$489$
$3,714$

Solution:

1.
Is $489$ a multiple of $2$ ?
Is the last digit $0, 2, 4, 6, \text{ or } 8$ ?	No.
	$489$ is not a multiple of $2$ .

2.
Is $3,714$ a multiple of $2$ ?
Is the last digit $0, 2, 4, 6, \text{ or } 8$ ?	Yes.
	$3,714$ is a multiple of $2$ .

try it

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

Multiples of $5$ between $1$ and $50$

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

Determine whether each of the following is a multiple of $5\text{:}$

$579$
$880$

Show Solution

Solution:

1.
Is $579$ a multiple of $5$ ?
Is the last digit $5\text{ or }0$ ?	No.
	$579$ is not a multiple of $5$ .

2.
Is $880$ a multiple of $5$ ?
Is the last digit $5\text{ or }0$ ?	Yes.
	$880$ is a multiple of $5$ .

try it

The table below highlights the multiples of $10$ between $1$ and $50$ . All multiples of $10$ all end with a zero.

Multiples of $10$ between $1$ and $50$

example

Determine whether each of the following is a multiple of $10\text{:}$

$425$
$350$

Show Solution

Solution:

1.
Is $425$ a multiple of $10$ ?
Is the last digit $0$ ?	No.
	$425$ is not a multiple of $10$ .

2.
Is $350$ a multiple of $10$ ?
Is the last digit $0$ ?	Yes.
	$350$ is a multiple of $10$ .

try it

The table below highlights multiples of $3$ . The pattern for multiples of $3$ is not as obvious as the patterns for multiples of $2,5,\text{and }10$ .

Multiples of $3$ between $1$ and $50$

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 the example below.

$\mathbf{\text{Multiple of 3}}$	$3$	$6$	$9$	$12$	$15$	$18$	$21$	$24$
$\mathbf{\text{Sum of digits}}$	$3$	$6$	$9$	$\begin{array}{c}\hfill 1+2\hfill \\ \hfill 3\hfill \end{array}$	$\begin{array}{c}\hfill 1+5\hfill \\ \hfill 6\hfill \end{array}$	$\begin{array}{c}\hfill 1+8\hfill \\ \hfill 9\hfill \end{array}$	$\begin{array}{c}\hfill 2+1\hfill \\ \hfill 3\hfill \end{array}$	$\begin{array}{c}\hfill 2+4\hfill \\ \hfill 6\hfill \end{array}$

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

Determine whether each of the given numbers is a multiple of $3\text{:}$

$645$
$10,519$

Show Solution

Solution:
1. 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 $645\text{ by }3$ .	$645\div 3$
The quotient is $215$ .	$3\cdot 215=645$

2. 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..	$645\div 3$
We can check this by dividing by $10,519\text{ by }3$ .	$\begin{array}{c}3,506\text{R}1\\ \hfill 3\overline{)10,519}\end{array}$

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 $3$ .

try it

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 $3$ .

The following video shows how to determine the first four multiples of 6.

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, $375\div 5$ is $75$ , so $375$ is $5\cdot 75$ . Notice in the last example 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 $3$ .

Divisibility

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

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

Divisibility Tests
A number is divisible by
$2$	if the last digit is $0,2,4,6,\text{or }8$
$3$	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

Determine whether $1,290$ is divisible by $2,3,5,\text{and }10$ .

Show Solution

Solution:
The table below 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.

Divisible by…?	Test	Divisible?	Check
$2$	Is last digit $0,2,4,6,\text{or }8?$ Yes.	yes	$1290\div 2=645$
$3$	$\text{Is sum of digits divisible by }3?$ $1+2+9+0=12$ 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 $2,3,5,\text{and }10$ .

try it

example

Determine whether $5,625$ is divisible by $2,3,5,\text{and }10$ .

Show Solution

Solution:
The table below applies the divisibility tests to $5,625$ and tests the results by finding the quotients.

Divisible by…?	Test	Divisible?	Check
$2$	Is last digit $0,2,4,6,\text{or }8?$ No.	no	$5625\div 2=2812.5$
$3$	$\text{Is sum of digits divisible by }3?$ $5+6+2+5=18$ 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 $10$ .

try it

The following video lesson shows how to determine whether a number is divisible by $2,3,4,5,6,8,9,10$

Module 1:The Language of Algebra