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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 =2232=49=36=2232=49=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 = 2315= 2 3 3 5 =2325 630 = 2 315 = 2 3 105 = 2 3 3 35 = 2 3 3 5 7 = 2 3 2 5 7 90 = 2 45 = 2315= 2 3 3 5 =2325 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 =23257=2957=630=23257=2957=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 =2511 484 = 2 242 = 2 2 121 = 2 2 11 11 = 2 2 11 2 . 33 = 3 11 110 = 2 55 =2511 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

Exercise 12

Exercise 13

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

Exercise 17

6 and 15

Exercise 18

Exercise 19

10 and 14

Exercise 20

Exercise 21

6 and 12

Exercise 22

Exercise 23

6 and 8

Exercise 24

Exercise 25

7 and 8

Exercise 26

Exercise 27

2 and 9

Exercise 28

Exercise 29

28 and 36

Exercise 30

Exercise 31

28 and 42

Exercise 32

Exercise 33

162 and 270

Exercise 34

Exercise 35

25 and 30

Exercise 36

Exercise 37

16 and 24

Exercise 38

Exercise 39

24 and 40

Exercise 40

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

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

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

848484848484 size 12{"84" cdot "84" cdot "84"} {}

Exercise 65

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

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