# 3.5 The least common multiple

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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

 Multiples of 2 Multiples of 3 Multiples of 8 Multiples of 10 $2×1=2$ $3×1=3$ $8×1=8$ $\text{10}×1=\text{10}$ $2×2=4$ $3×2=6$ $8×2=\text{16}$ $\text{10}×2=\text{20}$ $2×3=6$ $3×3=9$ $8×3=\text{24}$ $\text{10}×3=\text{30}$ $2×4=8$ $3×4=\text{12}$ $8×4=\text{32}$ $\text{10}×4=\text{40}$ $2×5=\text{10}$ $3×5=\text{15}$ $8×5=\text{40}$ $\text{10}×5=\text{50}$ $⋮$ $⋮$ $⋮$ $⋮$

## Practice set a

Find the first five multiples of the following numbers.

4

4, 8, 12, 16, 20

5

5, 10, 15, 20, 25

6

6, 12, 18, 24, 30

7

7, 14, 21, 28, 35

9

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

We can visualize common multiples using the number line. 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.

2 and 4

4, 8, 12, 16, 20

3 and 4

12, 24, 36, 48, 60

2 and 5

10, 20, 30, 40, 50

3 and 6

6, 12, 18, 24, 30

4 and 5

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 ( [link] ).

## 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.

9 and 12

1. $\begin{array}{c}9=3\cdot 3={3}^{2}\hfill \\ \text{12}=2\cdot 6=2\cdot 2\cdot 3={2}^{2}\cdot 3\hfill \end{array}$

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:

${2}^{2}$ from 12.

${3}^{2}$ from 9.

4. The LCM is the product of these numbers.

LCM $={2}^{2}\cdot {3}^{2}=4\cdot 9=\text{36}$

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

90 and 630

1. $\begin{array}{cccc}\mathrm{90}& =& 2\cdot \text{45}=2\cdot 3\cdot \mathrm{15}=2\cdot 3\cdot 3\cdot 5=2\cdot {3}^{2}\cdot 5\hfill & \\ \mathrm{630}& =& 2\cdot \text{315}=2\cdot 3\cdot \text{105}=2\cdot 3\cdot 3\cdot \text{35}\hfill & =2\cdot 3\cdot 3\cdot 5\cdot 7\\ & & & =2\cdot {3}^{2}\cdot 5\cdot 7\hfill \end{array}$

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:
• ${2}^{1}$ from either 90 or 630.
• ${3}^{2}$ from either 90 or 630.
• ${5}^{1}$ from either 90 or 630.
• ${7}^{1}$ from 630.
4. The LCM is the product of these numbers.

LCM $=2\cdot {3}^{2}\cdot 5\cdot 7=2\cdot 9\cdot 5\cdot 7=\text{630}$

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

33, 110, and 484

1. $\begin{array}{ccc}\mathrm{33}& =& 3\cdot \mathrm{11}\hfill \\ \mathrm{110}& =& 2\cdot \mathrm{55}=2\cdot 5\cdot \mathrm{11}\hfill \\ \mathrm{484}& =& 2\cdot \mathrm{242}=2\cdot 2\cdot \mathrm{121}=2\cdot 2\cdot \mathrm{11}\cdot \mathrm{11}={2}^{2}\cdot {\mathrm{11}}^{2}.\hfill \end{array}$

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:
• ${2}^{2}$ from 484.
• ${3}^{1}$ from 33.
• ${5}^{1}$ from 110
• ${\text{11}}^{2}$ from 484.
4. The LCM is the product of these numbers.

$\begin{array}{ccc}\text{LCM}& =& {2}^{2}\cdot 3\cdot 5\cdot {\mathrm{11}}^{2}\hfill \\ & =& 4\cdot 3\cdot 5\cdot \mathrm{121}\hfill \\ & =& \mathrm{7260}\hfill \end{array}$

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.

20 and 54

540

14 and 28

28

6 and 63

126

28, 40, and 98

1,960

16, 27, 125, and 363

6,534,000

## Exercises

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

8 and 12

24

6 and 15

8 and 10

40

10 and 14

4 and 6

12

6 and 12

9 and 18

18

6 and 8

5 and 6

30

7 and 8

3 and 4

12

2 and 9

7 and 9

63

28 and 36

24 and 36

72

28 and 42

240 and 360

720

162 and 270

20 and 24

120

25 and 30

24 and 54

216

16 and 24

36 and 48

144

24 and 40

15 and 21

105

50 and 140

7, 11, and 33

231

8, 10, and 15

18, 21, and 42

126

4, 5, and 21

45, 63, and 98

4,410

15, 25, and 40

12, 16, and 20

240

84 and 96

48 and 54

432

12, 16, and 24

12, 16, 24, and 36

144

6, 9, 12, and 18

8, 14, 28, and 32

224

18, 80, 108, and 490

22, 27, 130, and 225

193,050

38, 92, 115, and 189

8 and 8

8

12, 12, and 12

3, 9, 12, and 3

36

## Exercises for review

( [link] ) Round 434,892 to the nearest ten thousand.

( [link] ) How much bigger is 14,061 than 7,509?

6,552

( [link] ) Find the quotient. $\text{22},\text{428}÷\text{14}$ .

( [link] ) Expand ${\mathrm{84}}^{3}$ . Do not find the value.

$\text{84}\cdot \text{84}\cdot \text{84}$

( [link] ) Find the greatest common factor of 48 and 72.

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