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Multiplication and Division of Whole Numbers: Properties of Multiplication

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses properties of multiplication of whole numbers. By the end of the module students should be able to understand and appreciate the commutative and associative properties of multiplication and understand why 1 is the multiplicative identity.

Section Overview

  • The Commutative Property of Multiplication
  • The Associative Property of Multiplication
  • The Multiplicative Identity

We will now examine three simple but very important properties of multiplication.

The Commutative Property of Multiplication

Commutative Property of Multiplication

The product of two whole numbers is the same regardless of the order of the factors.

Sample Set A

Example 1

Multiply the two whole numbers.

6 and 7.

6 ⋅ 7 = 42 6 ⋅ 7 = 42 size 12{6 cdot 7="42"} {}

7 ⋅ 6 = 42 7 ⋅ 6 = 42 size 12{7 cdot 6="42"} {}

The numbers 6 and 7 can be multiplied in any order. Regardless of the order they are multiplied, the product is 42.

Practice Set A

Use the commutative property of multiplication to find the products in two ways.

Exercise 1

15 and 6.

Solution

15⋅6=9015⋅6=90 and 6⋅15=906⋅15=90

Exercise 2

432 and 428.

Solution

432⋅428=184,896432⋅428=184,896 and 428⋅432=184,896428⋅432=184,896

The Associative Property of Multiplication

Associative Property of Multiplication

If three whole numbers are multiplied, the product will be the same if the first two are multiplied first and then that product is multiplied by the third, or if the second two are multiplied first and that product is multiplied by the first. Note that the order of the factors is maintained.

It is a common mathematical practice to use parentheses to show which pair of numbers is to be combined first.

Sample Set B

Example 2

Multiply the whole numbers.

8, 3, and 14.

( 8 ⋅ 3 ) ⋅ 14 = 24 ⋅ 14 = 336 ( 8 ⋅ 3 ) ⋅ 14 = 24 ⋅ 14 = 336 size 12{ \( 8 cdot 3 \) cdot "14"="24" cdot "14"="336"} {}

8 ⋅ ( 3 ⋅ 14 ) = 8 ⋅ 42 = 336 8 ⋅ ( 3 ⋅ 14 ) = 8 ⋅ 42 = 336 size 12{8 cdot \( 3 cdot "14" \) =8 cdot "42"="336"} {}

Practice Set B

Use the associative property of multiplication to find the products in two ways.

Exercise 4

73, 18, and 126.

Solution

165,564

The Multiplicative Identity

The Multiplicative Identity is 1

The whole number 1 is called the multiplicative identity, since any whole num­ber multiplied by 1 is not changed.

Sample Set C

Example 3

Multiply the whole numbers.

12 and 1.

12 ⋅ 1 = 12 12 ⋅ 1 = 12 size 12{"12" cdot 1="12"} {}

1 ⋅ 12 = 12 1 ⋅ 12 = 12 size 12{1 cdot "12"="12"} {}

Practice Set C

Multiply the whole numbers.

Exercises

For the following problems, multiply the numbers.

Exercise 7

18 and 41.

Exercise 9

132 and 6.

Exercise 10

1000 and 326.

Solution

326,000

Exercise 11

70 and 1400.

Exercise 13

16, 40, and 5.

Exercise 14

10, 97, and 22.

Solution

21,340

Exercise 15

110, 0, and 85.

Exercise 16

1, 462, and 18.

Solution

8,316

Exercise 17

3,178, 101, and 5.

For the following 4 problems, show that the quantities yield the same products by performing the multiplications.

Exercise 18

(4⋅8)⋅2(4⋅8)⋅2 size 12{ \( 4 cdot 8 \) cdot 2} {} and 4⋅(8⋅2)4⋅(8⋅2) size 12{4 cdot \( 8 cdot 2 \) } {}

Solution

32⋅2=64=4⋅1632⋅2=64=4⋅16 size 12{"32" cdot 2="64"=4 cdot "16"} {}

Exercise 19

(100⋅62)⋅4(100⋅62)⋅4 size 12{ \( "100" cdot "62" \) cdot 4} {} and 100⋅(62⋅4)100⋅(62⋅4) size 12{"100" cdot \( "62" cdot 4 \) } {}

Exercise 20

23⋅(11⋅106)23⋅(11⋅106) size 12{"23" cdot \( "11" cdot "106" \) } {} and (23⋅11)⋅106(23⋅11)⋅106 size 12{ \( "23" cdot "11" \) cdot "106"} {}

Solution

23⋅1,166=26,818=253⋅10623⋅1,166=26,818=253⋅106 size 12{"23" cdot 1,"166"="26","818"="253" cdot "106"} {}

Exercise 21

1⋅(5⋅2)1⋅(5⋅2) size 12{1 cdot \( 5 cdot 2 \) } {} and (1⋅5)⋅2(1⋅5)⋅2 size 12{ \( 1 cdot 5 \) cdot 2} {}

Exercise 22

The fact that (a first number ⋅ a second number)⋅ a third number = a first number ⋅(a second number ⋅ a third number)(a first number ⋅ a second number)⋅ a third number = a first number ⋅(a second number ⋅ a third number) size 12{ \( "a first number " cdot " a second number" \) cdot " a third number "=" a first number " cdot \( "a second number " cdot " a third number" \) } {}is an example of the
property of mul­tiplication.

Solution

associative

Exercise 23

The fact that 1 ⋅ any number = that particular number1 ⋅ any number = that particular number size 12{"1 " cdot " any number "=" that particular number"} {}is an example of the
property of mul­tiplication.

Exercise 24

Use the numbers 7 and 9 to illustrate the com­mutative property of multiplication.

Solution

7 ⋅ 9 = 63 = 9 ⋅ 7 7 ⋅ 9 = 63 = 9 ⋅ 7 size 12{"7 " cdot " 9 "=" 63 "=" 9 " cdot " 7"} {}

Exercise 25

Use the numbers 6, 4, and 7 to illustrate the asso­ciative property of multiplication.

Exercises for Review

Exercise 26

((Reference)) In the number 84,526,098,441, how many millions are there?

Solution

6

Exercise 27

((Reference)) Replace the letter m with the whole number that makes the addition true. 85+  m̲9785+  m̲97alignr { stack { size 12{"85"} {} # size 12{ {underline {+m}} } {} # size 12{"97"} {} } } {}

Exercise 28

((Reference)) Use the numbers 4 and 15 to illustrate the commutative property of addition.

Solution

4 + 15 = 19 4 + 15 = 19 size 12{"4 "+" 15 "=" 19"} {}

15 + 4 = 19 15 + 4 = 19 size 12{"15 "+" 4 "=" 19"} {}

Exercise 29

((Reference)) Find the product. 8,000,000×1,0008,000,000×1,000 size 12{8,"000","000" times 1,"000"} {}.

Exercise 30

((Reference)) Specify which of the digits 2, 3, 4, 5, 6, 8,10 are divisors of the number 2,244.

Solution

2, 3, 4, 6

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