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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 size 12{2´1=2} {} 3 × 1 = 3 size 12{3´1=3} {} 8 × 1 = 8 size 12{8´1=8} {} 10 × 1 = 10 size 12{"10"´1="10"} {}
2 × 2 = 4 size 12{2´2=4} {} 3 × 2 = 6 size 12{3´2=6} {} 8 × 2 = 16 size 12{8´2="16"} {} 10 × 2 = 20 size 12{"10"´2="20"} {}
2 × 3 = 6 size 12{2´3=4} {} 3 × 3 = 9 size 12{3´3=9} {} 8 × 3 = 24 size 12{8´3="24"} {} 10 × 3 = 30 size 12{"10"´3="30"} {}
2 × 4 = 8 size 12{2´4=8} {} 3 × 4 = 12 size 12{3´4="12"} {} 8 × 4 = 32 size 12{8´4="32"} {} 10 × 4 = 40 size 12{"10"´4="40"} {}
2 × 5 = 10 size 12{2´5="10"} {} 3 × 5 = 15 size 12{3´5="15"} {} 8 × 5 = 40 size 12{8´5="40"} {} 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.

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.

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.

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Practice set b

Find the first five common multiples of the following numbers.

2 and 4

4, 8, 12, 16, 20

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3 and 4

12, 24, 36, 48, 60

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2 and 5

10, 20, 30, 40, 50

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3 and 6

6, 12, 18, 24, 30

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4 and 5

20, 40, 60, 80, 100

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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 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.

9 and 12

  1. 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:

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

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

  4. The LCM is the product of these numbers.

    LCM = 2 2 3 2 = 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.

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90 and 630

  1. 90 = 2 45 = 2 3 15 = 2 3 3 5 = 2 3 2 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:
    • 2 1 size 12{2 rSup { size 8{1} } } {} from either 90 or 630.
    • 3 2 size 12{3 rSup { size 8{2} } } {} from either 90 or 630.
    • 5 1 size 12{5 rSup { size 8{1} } } {} from either 90 or 630.
    • 7 1 size 12{7 rSup { size 8{1} } } {} from 630.
  4. The LCM is the product of these numbers.

    LCM = 2 3 2 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.

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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 .

  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 size 12{2 rSup { size 8{2} } } {} from 484.
    • 3 1 size 12{3 rSup { size 8{1} } } {} from 33.
    • 5 1 size 12{5 rSup { size 8{1} } } {} from 110
    • 11 2 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

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

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Practice set c

Find the LCM of the following numbers.

16, 27, 125, and 363

6,534,000

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Exercises

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

22, 27, 130, and 225

193,050

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Exercises for review

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

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

6,552

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( [link] ) Find the quotient. 22 , 428 ÷ 14 size 12{"22","428"¸"14"} {} .

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( [link] ) Expand 84 3 . Do not find the value.

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

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

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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