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By the end of this section, you will be able to:
  • Find the prime factorization of a composite number
  • Find the least common multiple (LCM) of two numbers

Before you get started, take this readiness quiz.

  1. Is 810 divisible by 2 , 3 , 5 , 6 , or 10 ?
    If you missed this problem, review Find Multiples and Factors .
  2. Is 127 prime or composite?
    If you missed this problem, review Find Multiples and Factors .
  3. Write 2 2 2 2 in exponential notation.
    If you missed this problem, review Use the Language of Algebra .

Find the prime factorization of a composite number

In the previous section, we found the factors of a number. Prime numbers have only two factors, the number 1 and the prime number itself. Composite numbers have more than two factors, and every composite number can be written as a unique product of primes. This is called the prime factorization    of a number. When we write the prime factorization of a number, we are rewriting the number as a product of primes. Finding the prime factorization of a composite number will help you later in this course.

Prime factorization

The prime factorization of a number is the product of prime numbers that equals the number.

Doing the Manipulative Mathematics activity “Prime Numbers” will help you develop a better sense of prime numbers.

You may want to refer to the following list of prime numbers less than 50 as you work through this section.

2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47

Prime factorization using the factor tree method

One way to find the prime factorization of a number is to make a factor tree . We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment—a “branch” of the factor tree.

If a factor is prime, we circle it (like a bud on a tree), and do not factor that “branch” any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree.

We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization.

For example, let’s find the prime factorization of 36 . We can start with any factor pair such as 3 and 12 . We write 3 and 12 below 36 with branches connecting them.

The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end.

The factor 3 is prime, so we circle it. The factor 12 is composite, so we need to find its factors. Let’s use 3 and 4 . We write these factors on the tree under the 12 .

The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end.

The factor 3 is prime, so we circle it. The factor 4 is composite, and it factors into 2 · 2 . We write these factors under the 4 . Since 2 is prime, we circle both 2 s .

The figure shows a factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end with a circle around it. Two more branches are splitting out from under 4. Both the left and right branch have the number 2 at the end with a circle around it.

The prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.

2 2 3 3

In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.

2 2 3 3 2 2 3 2

Note that we could have started our factor tree with any factor pair of 36 . We chose 12 and 3 , but the same result would have been the same if we had started with 2 and 18 , 4 and 9 , or 6 and 6 .

Questions & Answers

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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Kimberly Reply
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.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
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)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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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