# 2.5 Prime factorization and the least common multiple  (Page 3/8)

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Find the prime factorization using the ladder method: $80$

2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5, or 2 4 ⋅ 5

Find the prime factorization using the ladder method: $60$

2 ⋅ 2 ⋅ 3 ⋅ 5, or 2 2 ⋅ 3 ⋅ 5

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. $2\cdot 2\cdot 2\cdot 2\cdot 3$ ${2}^{4}\cdot 3$

Find the prime factorization using the ladder method. $126$

2 ⋅ 3 ⋅ 3 ⋅ 7, or 2 ⋅ 3 2 ⋅ 7

Find the prime factorization using the ladder method. $294$

2 ⋅ 3 ⋅ 7 ⋅ 7, or 2 ⋅ 3 ⋅ 7 2

## 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.

$\begin{array}{c}10\text{:}10,20,30,40,\phantom{\rule{0.2em}{0ex}}50,60,70,80,90,100,110,\text{…}\hfill \\ 25\text{:}25,\phantom{\rule{0.2em}{0ex}}50,75,\phantom{\rule{0.2em}{0ex}}100,125,\text{…}\hfill \end{array}$

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.

$\begin{array}{l}\text{15:}\phantom{\rule{0.2em}{0ex}}15,30,45,\phantom{\rule{0.2em}{0ex}}60,75,90,105,120\hfill \\ \text{20:}\phantom{\rule{0.2em}{0ex}}20,40,\phantom{\rule{0.2em}{0ex}}60,80,100,120,140,160\hfill \end{array}$

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.

Find the least common multiple (LCM) of the given numbers: $9\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}12$

36

Find the least common multiple (LCM) of the given numbers: $18\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}24$

72

## 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.

$12=2\cdot 2\cdot 3\phantom{\rule{3em}{0ex}}18=2\cdot 3\cdot 3$

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

$\begin{array}{l}12=2\cdot 2\cdot 3\hfill \\ 18=2\cdot \phantom{\rule{1.1em}{0ex}}3\cdot 3\end{array}$

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

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.

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absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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SUYASH
What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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