Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Derived copy of Fundamentals of Mathematics » The Least Common Multiple

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Endorsed by Endorsed (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
  • CCQ display tagshide tags

    This module is included in aLens by: Community College of QatarAs a part of collection: "Fundamentals of Mathematics"

    Comments:

    "Used as supplemental materials for developmental math courses."

    Click the "CCQ" link to see all content they endorse.

    Click the tag icon tag icon to display tags associated with this content.

  • College Open Textbooks display tagshide tags

    This module is included inLens: Community College Open Textbook Collaborative
    By: CC Open Textbook CollaborativeAs a part of collection: "Fundamentals of Mathematics"

    Comments:

    "Reviewer's Comments: 'I would recommend this text for a basic math course for students moving on to elementary algebra. The information in most chapters is useful, very clear, and easily […]"

    Click the "College Open Textbooks" link to see all content they endorse.

    Click the tag icon tag icon to display tags associated with this content.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Featured Content display tagshide tags

    This module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection: "Fundamentals of Mathematics"

    Comments:

    "Fundamentals of Mathematics is a work text that covers the traditional topics studied in a modern prealgebra course, as well as topics of estimation, elementary analytic geometry, and […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • UniqU content

    This module is included inLens: UniqU's lens
    By: UniqU, LLCAs a part of collection: "Fundamentals of Mathematics"

    Click the "UniqU content" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

The Least Common Multiple

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 the least common multiple. By the end of the module students should be able to find the least common multiple of two or more whole numbers.

Section Overview

  • Multiples
  • Common Multiples
  • The Least Common Multiple (LCM)
  • Finding the Least Common Multiple

Multiples

When a whole number is multiplied by other whole numbers, with the exception of zero, the resulting products are called multiples of the given whole number. Note that any whole number is a multiple of itself.

Sample Set A

Table 1
Multiples of 2 Multiples of 3 Multiples of 8 Multiples of 10
2 × 1 = 2 2 × 1 = 2 size 12{2´1=2} {} 3 × 1 = 3 3 × 1 = 3 size 12{3´1=3} {} 8 × 1 = 8 8 × 1 = 8 size 12{8´1=8} {} 10 × 1 = 10 10 × 1 = 10 size 12{"10"´1="10"} {}
2 × 2 = 4 2 × 2 = 4 size 12{2´2=4} {} 3 × 2 = 6 3 × 2 = 6 size 12{3´2=6} {} 8 × 2 = 16 8 × 2 = 16 size 12{8´2="16"} {} 10 × 2 = 20 10 × 2 = 20 size 12{"10"´2="20"} {}
2 × 3 = 6 2 × 3 = 6 size 12{2´3=4} {} 3 × 3 = 9 3 × 3 = 9 size 12{3´3=9} {} 8 × 3 = 24 8 × 3 = 24 size 12{8´3="24"} {} 10 × 3 = 30 10 × 3 = 30 size 12{"10"´3="30"} {}
2 × 4 = 8 2 × 4 = 8 size 12{2´4=8} {} 3 × 4 = 12 3 × 4 = 12 size 12{3´4="12"} {} 8 × 4 = 32 8 × 4 = 32 size 12{8´4="32"} {} 10 × 4 = 40 10 × 4 = 40 size 12{"10"´4="40"} {}
2 × 5 = 10 2 × 5 = 10 size 12{2´5="10"} {} 3 × 5 = 15 3 × 5 = 15 size 12{3´5="15"} {} 8 × 5 = 40 8 × 5 = 40 size 12{8´5="40"} {} 10 × 5 = 50 10 × 5 = 50 size 12{"10"´5="50"} {}
⋮ ⋮ size 12{ dotsvert } {} ⋮ ⋮ size 12{ dotsvert } {} ⋮ ⋮ size 12{ dotsvert } {} ⋮ ⋮ size 12{ dotsvert } {}

Practice Set A

Find the first five multiples of the following numbers.

Exercise 1

4

Solution

4, 8, 12, 16, 20

Exercise 2

5

Solution

5, 10, 15, 20, 25

Exercise 3

6

Solution

6, 12, 18, 24, 30

Exercise 4

7

Solution

7, 14, 21, 28, 35

Exercise 5

9

Solution

9, 18, 27, 36, 45

Common Multiples

There will be times when we are given two or more whole numbers and we will need to know if there are any multiples that are common to each of them. If there are, we will need to know what they are. For example, some of the multiples that are common to 2 and 3 are 6, 12, and 18.

Sample Set B

Example 1

We can visualize common multiples using the number line.

A number line. On the top are lines connecting every second number from 2 to 18. This part is labeled, multiples of 2. On the bottom are lines connecting every third number from 3 to 18. This part is labeled, multiples of 3. Sometimes, the lines land on the same number. This happens on 6, 12, and 18, which are labeled, first, second, and third common multiple, respectively.

Notice that the common multiples can be divided by both whole numbers.

Practice Set B

Find the first five common multiples of the following numbers.

Exercise 6

2 and 4

Solution

4, 8, 12, 16, 20

Exercise 7

3 and 4

Solution

12, 24, 36, 48, 60

Exercise 8

2 and 5

Solution

10, 20, 30, 40, 50

Exercise 9

3 and 6

Solution

6, 12, 18, 24, 30

Exercise 10

4 and 5

Solution

20, 40, 60, 80, 100

The Least Common Multiple (LCM)

Notice that in our number line visualization of common multiples (above), the first common multiple is also the smallest, or least common multiple, abbreviated by LCM.

Least Common Multiple

The least common multiple, LCM, of two or more whole numbers is the smallest whole number that each of the given numbers will divide into without a remainder.

The least common multiple will be extremely useful in working with fractions ((Reference)).

Finding the Least Common Multiple

Finding the LCM

To find the LCM of two or more numbers:

  1. Write the prime factorization of each number, using exponents on repeated factors.
  2. Write each base that appears in each of the prime factorizations.
  3. To each base, attach the largest exponent that appears on it in the prime factorizations.
  4. The LCM is the product of the numbers found in step 3.

There are some major differences between using the processes for obtaining the GCF and the LCM that we must note carefully:

The Difference Between the Processes for Obtaining the GCF and the LCM

  1. Notice the difference between step 2 for the LCM and step 2 for the GCF. For the GCF, we use only the bases that are common in the prime factorizations, whereas for the LCM, we use each base that appears in the prime factorizations.
  2. Notice the difference between step 3 for the LCM and step 3 for the GCF. For the GCF, we attach the smallest exponents to the common bases, whereas for the LCM, we attach the largest exponents to the bases.

Sample Set C

Find the LCM of the following numbers.

Example 2

9 and 12

  1. 9 = 3 ⋅ 3 = 3 2 12 = 2 ⋅ 6 = 2 ⋅ 2 ⋅ 3 = 2 2 ⋅ 3 9 = 3 ⋅ 3 = 3 2 size 12{9=3 cdot 3=3 rSup { size 8{2} } } {} 12 = 2 ⋅ 6 = 2 ⋅ 2 ⋅ 3 = 2 2 ⋅ 3 size 12{"12"=2 cdot 6=2 cdot 2 cdot 3=2 rSup { size 8{2} } cdot 3} {}

  2. The bases that appear in the prime factorizations are 2 and 3.
  3. The largest exponents appearing on 2 and 3 in the prime factorizations are, respectively, 2 and 2:

    2222 size 12{2 rSup { size 8{2} } } {} from 12.

    3232 size 12{3 rSup { size 8{2} } } {} from 9.

  4. The LCM is the product of these numbers.

    LCM =22⋅32=4⋅9=36=22⋅32=4⋅9=36 size 12{ {}=2 rSup { size 8{2} } cdot 3 rSup { size 8{2} } =4 cdot 9="36"} {}

Thus, 36 is the smallest number that both 9 and 12 divide into without remainders.

Example 3

90 and 630

  1. 90 = 2 ⋅ 45 = 2⋅3⋅15= 2 ⋅ 3 ⋅ 3 ⋅ 5 =2⋅32⋅5 630 = 2 ⋅ 315 = 2 ⋅ 3 ⋅ 105 = 2 ⋅ 3 ⋅ 3 ⋅ 35 = 2 ⋅ 3 ⋅ 3 ⋅ 5 ⋅ 7 = 2 ⋅ 3 2 ⋅ 5 ⋅ 7 90 = 2 ⋅ 45 = 2⋅3⋅15= 2 ⋅ 3 ⋅ 3 ⋅ 5 =2⋅32⋅5 630 = 2 ⋅ 315 = 2 ⋅ 3 ⋅ 105 = 2 ⋅ 3 ⋅ 3 ⋅ 35 = 2 ⋅ 3 ⋅ 3 ⋅ 5 ⋅ 7 = 2 ⋅ 3 2 ⋅ 5 ⋅ 7

  2. The bases that appear in the prime factorizations are 2, 3, 5, and 7.
  3. The largest exponents that appear on 2, 3, 5, and 7 are, respectively, 1, 2, 1, and 1:
    • 2121 size 12{2 rSup { size 8{1} } } {} from either 90 or 630.
    • 3232 size 12{3 rSup { size 8{2} } } {} from either 90 or 630.
    • 5151 size 12{5 rSup { size 8{1} } } {} from either 90 or 630.
    • 7171 size 12{7 rSup { size 8{1} } } {} from 630.
  4. The LCM is the product of these numbers.

    LCM =2⋅32⋅5⋅7=2⋅9⋅5⋅7=630=2⋅32⋅5⋅7=2⋅9⋅5⋅7=630 size 12{ {}=2 cdot 3 rSup { size 8{2} } cdot 5 cdot 7=2 cdot 9 cdot 5 cdot 7="630"} {}

Thus, 630 is the smallest number that both 90 and 630 divide into with no remainders.

Example 4

33, 110, and 484

  1. 33 = 3 ⋅ 11 110 = 2 ⋅ 55 =2⋅5⋅11 484 = 2 ⋅ 242 = 2 ⋅ 2 ⋅ 121 = 2 ⋅ 2 ⋅ 11 ⋅ 11 = 2 2 ⋅ 11 2 . 33 = 3 ⋅ 11 110 = 2 ⋅ 55 =2⋅5⋅11 484 = 2 ⋅ 242 = 2 ⋅ 2 ⋅ 121 = 2 ⋅ 2 ⋅ 11 ⋅ 11 = 2 2 ⋅ 11 2 .

  2. The bases that appear in the prime factorizations are 2, 3, 5, and 11.
  3. The largest exponents that appear on 2, 3, 5, and 11 are, respectively, 2, 1, 1, and 2:
    • 2222 size 12{2 rSup { size 8{2} } } {} from 484.
    • 3131 size 12{3 rSup { size 8{1} } } {} from 33.
    • 5151 size 12{5 rSup { size 8{1} } } {} from 110
    • 112112 size 12{"11" rSup { size 8{2} } } {} from 484.
  4. The LCM is the product of these numbers.

    LCM = 2 2 ⋅ 3 ⋅ 5 ⋅ 11 2 = 4 ⋅ 3 ⋅ 5 ⋅ 121 = 7260 LCM = 2 2 ⋅ 3 ⋅ 5 ⋅ 11 2 = 4 ⋅ 3 ⋅ 5 ⋅ 121 = 7260

    Thus, 7260 is the smallest number that 33, 110, and 484 divide into without remainders.

Practice Set C

Find the LCM of the following numbers.

Exercise 11

20 and 54

Solution

540

Exercise 12

14 and 28

Solution

28

Exercise 13

6 and 63

Solution

126

Exercise 14

28, 40, and 98

Solution

1,960

Exercise 15

16, 27, 125, and 363

Solution

6,534,000

Exercises

For the following problems, find the least common multiple of the numbers.

Exercise 16

8 and 12

Solution

24

Exercise 17

6 and 15

Exercise 18

8 and 10

Solution

40

Exercise 19

10 and 14

Exercise 20

4 and 6

Solution

12

Exercise 21

6 and 12

Exercise 22

9 and 18

Solution

18

Exercise 23

6 and 8

Exercise 24

5 and 6

Solution

30

Exercise 25

7 and 8

Exercise 26

3 and 4

Solution

12

Exercise 27

2 and 9

Exercise 28

7 and 9

Solution

63

Exercise 29

28 and 36

Exercise 30

24 and 36

Solution

72

Exercise 31

28 and 42

Exercise 32

240 and 360

Solution

720

Exercise 33

162 and 270

Exercise 34

20 and 24

Solution

120

Exercise 35

25 and 30

Exercise 36

24 and 54

Solution

216

Exercise 37

16 and 24

Exercise 38

36 and 48

Solution

144

Exercise 39

24 and 40

Exercise 40

15 and 21

Solution

105

Exercise 41

50 and 140

Exercise 42

7, 11, and 33

Solution

231

Exercise 43

8, 10, and 15

Exercise 44

18, 21, and 42

Solution

126

Exercise 45

4, 5, and 21

Exercise 46

45, 63, and 98

Solution

4,410

Exercise 47

15, 25, and 40

Exercise 48

12, 16, and 20

Solution

240

Exercise 49

84 and 96

Exercise 50

48 and 54

Solution

432

Exercise 51

12, 16, and 24

Exercise 52

12, 16, 24, and 36

Solution

144

Exercise 53

6, 9, 12, and 18

Exercise 54

8, 14, 28, and 32

Solution

224

Exercise 55

18, 80, 108, and 490

Exercise 56

22, 27, 130, and 225

Solution

193,050

Exercise 57

38, 92, 115, and 189

Exercise 58

8 and 8

Solution

8

Exercise 59

12, 12, and 12

Exercise 60

3, 9, 12, and 3

Solution

36

Exercises for Review

Exercise 61

((Reference)) Round 434,892 to the nearest ten thousand.

Exercise 62

((Reference)) How much bigger is 14,061 than 7,509?

Solution

6,552

Exercise 63

((Reference)) Find the quotient. 22,428÷1422,428÷14 size 12{"22","428"¸"14"} {}.

Exercise 64

((Reference)) Expand 843843. Do not find the value.

Solution

84⋅84⋅8484⋅84⋅84 size 12{"84" cdot "84" cdot "84"} {}

Exercise 65

((Reference)) Find the greatest common factor of 48 and 72.

Collection Navigation

Content actions

Download module as:

PDF | More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks