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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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Questions & Answers

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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
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Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
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is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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What is meant by 'nano scale'?
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scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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what is Nano technology ?
Bob Reply
write examples of Nano molecule?
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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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why?
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what school?
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biomolecules are e building blocks of every organics and inorganic materials.
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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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