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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses prime factorization of natural numbers. By the end of the module students should be able to determine the factors of a whole number, distinguish between prime and composite numbers, be familiar with the fundamental principle of arithmetic and find the prime factorization of a whole number.

Section overview

  • Factors
  • Determining the Factors of a Whole Number
  • Prime and Composite Numbers
  • The Fundamental Principle of Arithmetic
  • The Prime Factorization of a Natural Number

Factors

From observations made in the process of multiplication, we have seen that

( factor ) ( factor ) = product size 12{ \( "factor" \) cdot " " \( "factor" \) =" product"} {}

Factors, product

The two numbers being multiplied are the factors and the result of the multiplication is the product . Now, using our knowledge of division, we can see that a first number is a factor of a second number if the first number divides into the second number a whole number of times (without a remainder).

One number as a factor of another

A first number is a factor of a second number if the first number divides into the second number a whole number of times (without a remainder).

We show this in the following examples:

3 is a factor of 27, since 27 ÷ 3 = 9 size 12{"27"÷3=9} {} , or 3 9 = 27 size 12{3 cdot 9="27"} {} .

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7 is a factor of 56, since 56 ÷ 7 = 8 size 12{"56"÷7=8} {} , or 7 8 = 56 size 12{7 cdot 8="56"} {} .

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4 is not a factor of 10, since 10 ÷ 4 = 2R2 size 12{"10"÷4=2R2} {} . (There is a remainder.)

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Determining the factors of a whole number

We can use the tests for divisibility from [link] to determine all the factors of a whole number.

Sample set a

Find all the factors of 24.

Try 1: 24 ÷ 1 = 24 size 12{"24"÷1="24"} {} 1 and 24 are factors Try 2: 24 is even, so 24 is divisible by 2. 24 ÷ 2 = 12 size 12{"24"÷2="12"} {} 2 and 12 are factors Try 3: 2 + 4 = 6 size 12{2+4=6} {} and 6 is divisible by 3, so 24 is divisible by 3. 24 ÷ 3 = 8 size 12{"24"÷3=8} {} 3 and 8 are factors Try 4: 24 ÷ 4 = 6 size 12{"24"÷4=6} {} 4 and 6 are factors Try 5: 24 ÷ 5 = 4R4 size 12{"24"÷5=4R4} {} 5  is not a factor.

The next number to try is 6, but we already have that 6 is a factor. Once we come upon a factor that we already have discovered, we can stop.

All the whole number factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

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Practice set a

Find all the factors of each of the following numbers.

Prime and composite numbers

Notice that the only factors of 7 are 1 and 7 itself, and that the only factors of 3 are 1 and 3 itself. However, the number 8 has the factors 1, 2, 4, and 8, and the number 10 has the factors 1, 2, 5, and 10. Thus, we can see that a whole number can have only two factors (itself and 1) and another whole number can have several factors.

We can use this observation to make a useful classification for whole numbers: prime numbers and composite numbers.

Prime number

A whole number (greater than one) whose only factors are itself and 1 is called a prime number .

The number 1 is Not A prime number

The first seven prime numbers are 2, 3, 5, 7, 11, 13, and 17. Notice that the whole number 1 is not considered to be a prime number, and the whole number 2 is the first prime and the only even prime number.

Composite number

A whole number composed of factors other than itself and 1 is called a compos­ite number . Composite numbers are not prime numbers.

Questions & Answers

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
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Google
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
Hafiz Reply
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
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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In the number 779,844,205 how many ten millions are there?
TELLY Reply
From 1973 to 1979, in the United States, there was an increase of 166.6% of Ph.D. social scien­tists to 52,000. How many were there in 1973?
Khizar Reply
7hours 36 min - 4hours 50 min
Tanis Reply

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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