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Find the prime factorization using the ladder method: 80

2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5, or 2 4 ⋅ 5

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Find the prime factorization using the ladder method: 60

2 ⋅ 2 ⋅ 3 ⋅ 5, or 2 2 ⋅ 3 ⋅ 5

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Find the prime factorization of 48 using the ladder method.

Solution

Divide the number by the smallest prime, 2. .
Continue dividing by 2 until it no longer divides evenly. .
The quotient, 3, is prime, so the ladder is complete. Write the prime factorization of 48. 2 2 2 2 3
2 4 3
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Find the prime factorization using the ladder method. 126

2 ⋅ 3 ⋅ 3 ⋅ 7, or 2 ⋅ 3 2 ⋅ 7

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Find the prime factorization using the ladder method. 294

2 ⋅ 3 ⋅ 7 ⋅ 7, or 2 ⋅ 3 ⋅ 7 2

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Find the least common multiple (lcm) of two numbers

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.

Listing multiples method

A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of 10 and 25 . We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples.

10 : 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 , 110 , 25 : 25 , 50 , 75 , 100 , 125 ,

We see that 50 and 100 appear in both lists. They are common multiples of 10 and 25 . We would find more common multiples if we continued the list of multiples for each.

The smallest number that is a multiple of two numbers is called the least common multiple    (LCM). So the least LCM of 10 and 25 is 50 .

Find the least common multiple (lcm) of two numbers by listing multiples.

  1. List the first several multiples of each number.
  2. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
  3. Look for the smallest number that is common to both lists.
  4. This number is the LCM.

Find the LCM of 15 and 20 by listing multiples.

Solution

List the first several multiples of 15 and of 20 . Identify the first common multiple.

15: 15 , 30 , 45 , 60 , 75 , 90 , 105 , 120 20: 20 , 40 , 60 , 80 , 100 , 120 , 140 , 160

The smallest number to appear on both lists is 60 , so 60 is the least common multiple of 15 and 20 .

Notice that 120 is on both lists, too. It is a common multiple, but it is not the least common multiple.

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Find the least common multiple (LCM) of the given numbers: 9 and 12

36

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Find the least common multiple (LCM) of the given numbers: 18 and 24

72

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Prime factors method

Another way to find the least common multiple of two numbers is to use their prime factors. We’ll use this method to find the LCM of 12 and 18 .

We start by finding the prime factorization of each number.

12 = 2 2 3 18 = 2 3 3

Then we write each number as a product of primes, matching primes vertically when possible.

12 = 2 2 3 18 = 2 3 3

Now we bring down the primes in each column. The LCM is the product of these factors.

The image shows the prime factorization of 12 written as the equation 12 equals 2 times 2 times 3. Below this equation is another showing the prime factorization of 18 written as the equation 18 equals 2 times 3 times 3. The two equations line up vertically at the equal symbol. The first 2 in the prime factorization of 12 aligns with the 2 in the prime factorization of 18. Under the second 2 in the prime factorization of 12 is a gap in the prime factorization of 18. Under the 3 in the prime factorization of 12 is the first 3 in the prime factorization of 18. The second 3 in the prime factorization has no factors above it from the prime factorization of 12. A horizontal line is drawn under the prime factorization of 18. Below this line is the equation LCM equal to 2 times 2 times 3 times 3. Arrows are drawn down vertically from the prime factorization of 12 through the prime factorization of 18 ending at the LCM equation. The first arrow starts at the first 2 in the prime factorization of 12 and continues down through the 2 in the prime factorization of 18. Ending with the first 2 in the LCM. The second arrow starts at the next 2 in the prime factorization of 12 and continues down through the gap in the prime factorization of 18. Ending with the second 2 in the LCM. The third arrow starts at the 3 in the prime factorization of 12 and continues down through the first 3 in the prime factorization of 18. Ending with the first 3 in the LCM. The last arrow starts at the second 3 in the prime factorization of 18 and points down to the second 3 in the LCM.

Notice that the prime factors of 12 and the prime factors of 18 are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that 36 is the least common multiple.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Practice Key Terms 2

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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