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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 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

Solution

156=90156=90 and 615=90615=90

Exercise 2

Solution

432428=184,896432428=184,896 and 428432=184,896428432=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 ) 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.

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 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 17

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

Exercise 18

(48)2(48)2 size 12{ $$4 cdot 8$$ cdot 2} {} and 4(82)4(82) size 12{4 cdot $$8 cdot 2$$ } {}

Solution

322=64=416322=64=416 size 12{"32" cdot 2="64"=4 cdot "16"} {}

Exercise 19

(10062)4(10062)4 size 12{ $$"100" cdot "62"$$ cdot 4} {} and 100(624)100(624) size 12{"100" cdot $$"62" cdot 4$$ } {}

Exercise 20

23(11106)23(11106) size 12{"23" cdot $$"11" cdot "106"$$ } {} and (2311)106(2311)106 size 12{ $$"23" cdot "11"$$ cdot "106"} {}

Solution

231,166=26,818=253106231,166=26,818=253106 size 12{"23" cdot 1,"166"="26","818"="253" cdot "106"} {}

Exercise 21

1(52)1(52) size 12{1 cdot $$5 cdot 2$$ } {} and (15)2(15)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.

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?

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.

2, 3, 4, 6

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