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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.

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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.

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.

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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 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.

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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 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.

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Exercises

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

5, 6

2 · 3 · 5

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28, 36

2 2 · 3 2 · 7

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28, 42

2 2 · 3 · 7

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25, 30

2 · 3 · 5 2

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15, 21

3 · 5 · 7

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8, 10, 15

2 3 · 3 · 5

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45, 63, 98

2 · 3 2 · 5 · 7 2

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12, 16, 20

2 4 · 3 · 5

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12, 16, 24, 36

2 4 · 3 2

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8, 14, 28, 32

2 5 · 7

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Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
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Commplementary angles
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
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silver nanoparticles could handle the job?
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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