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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. Therefore, no work space is provided, nor does the chapter contain all of the pedagogical features of the text. As a review, this chapter can be assigned at the discretion of the instructor and can also be a valuable reference tool for the student.

Overview

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

Multiples

Multiples

When a whole number is multiplied by other whole numbers, with the exception of Multiples zero, the resulting products are called multiples of the given whole number.

Multiples of 2 Multiples of 3 Multiples of 8 Multiples of 10
2 · 1 = 2 3 · 1 = 3 8 · 1 = 8 10 · 1 = 10
2 · 2 = 4 3 · 2 = 6 8 · 2 = 16 10 · 2 = 20
2 · 3 = 6 3 · 3 = 9 8 · 3 = 24 10 · 3 = 30
2 · 4 = 8 3 · 4 = 12 8 · 4 = 32 10 · 4 = 40
2 · 5 = 10 3 · 5 = 15 8 · 5 = 40 10 · 5 = 50

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 a

We can visualize common multiples using the number line.

A horizontal line numbered from zero to eighteen. Multiples of two and three are marked with dark circles, and are connected through arcs. Six, twelve and eighteen are labeled as

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

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.

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.

Sample set b

Find the LCM of the following number.

 9 and 12

  1. 9 = 3 · 3 = 3 2 12 = 2 · 6 = 2 · 2 · 3 = 2 2 · 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 (or 2 2 from 12, and 3 2 from 9).
  4. The LCM is the product of these numbers.

    LCM  = 2 2 · 3 2 = 4 · 9 = 36
 Thus, 36 is the smallest number that both 9 and 12 divide into without remainders.

 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 from either 9 0  or 63 0 3 2 from either 9 0  or 63 0 5 1 from either 9 0  or 63 0 7 1 from 63 0
  4. The LCM is the product of these numbers.

    LCM  = 2 · 3 2 · 5 · 7 = 2 · 9 · 5 · 7 = 630
 Thus, 630 is the smallest number that both 90 and 630 divide into with no remainders.

 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 from  484 3 1 from  33 5 1 from  110 11 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.

Exercises

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

8, 12

2 3 · 3

8, 10

6, 12

2 2 · 3

9, 18

5, 6

2 · 3 · 5

7, 9

28, 36

2 2 · 3 2 · 7

24, 36

28, 42

2 2 · 3 · 7

20, 24

25, 30

2 · 3 · 5 2

24, 54

16, 24

2 4 · 3

36, 48

15, 21

3 · 5 · 7

7, 11, 33

8, 10, 15

2 3 · 3 · 5

4, 5, 21

45, 63, 98

2 · 3 2 · 5 · 7 2

15, 25, 40

12, 16, 20

2 4 · 3 · 5

12, 16, 24

12, 16, 24, 36

2 4 · 3 2

6, 9, 12, 18

8, 14, 28, 32

2 5 · 7

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
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
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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