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

 Page 3 / 8

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.

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?