2.4 Find multiples and factors

Prealgebra Page 1 / 9

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

Before you get started, take this readiness quiz.

Which of the following numbers are counting numbers (natural numbers)?
$0, 4, 215$
If you missed this problem, review Introduction to Whole Numbers .
Find the sum of $3, 5,$ and $7 .$
If you missed the problem, review Use the Language of Algebra .

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. [link] shows the multiples of $2$ through $9$ for the first twelve counting numbers.

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

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 $2, 5, 10, and 3$ 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.

[link] 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. — Multiples of $2$ between $1$ and $50$

The last digit of each highlighted number in [link] 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 $2 .$

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

ⓐ $489$
ⓑ $3,714$

Solution

ⓐ
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.

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

ⓐ $678$
ⓑ $21,493$

ⓐ yes
ⓑ no

Got questions? Get instant answers now!

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

ⓐ $979$
ⓑ $17,780$

ⓐ no
ⓑ yes

Got questions? Get instant answers now!

Now let’s look at multiples of $5 .$ [link] 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.

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

ⓐ $579$
ⓑ $880$

Solution

ⓐ
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.

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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