<< Chapter < Page Chapter >> Page >
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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

 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.

Got questions? Get instant answers now!

 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.

Got questions? Get instant answers now!

Exercises

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

5, 6

2 · 3 · 5

Got questions? Get instant answers now!

28, 36

2 2 · 3 2 · 7

Got questions? Get instant answers now!

28, 42

2 2 · 3 · 7

Got questions? Get instant answers now!

25, 30

2 · 3 · 5 2

Got questions? Get instant answers now!

15, 21

3 · 5 · 7

Got questions? Get instant answers now!

8, 10, 15

2 3 · 3 · 5

Got questions? Get instant answers now!

45, 63, 98

2 · 3 2 · 5 · 7 2

Got questions? Get instant answers now!

12, 16, 20

2 4 · 3 · 5

Got questions? Get instant answers now!

12, 16, 24, 36

2 4 · 3 2

Got questions? Get instant answers now!

8, 14, 28, 32

2 5 · 7

Got questions? Get instant answers now!

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask