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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. Therefore, no work space is provided, nor does the chapter contain all of the pedagogical features of the text. As a review, this chapter can be assigned at the discretion of the instructor and can also be a valuable reference tool for the student.

Overview

  • Prime And Composite Numbers
  • The Fundamental Principle Of Arithmetic
  • The Prime Factorization Of A Whole Number

Prime and composite numbers

Notice that the only factors of 7 are 1 and 7 itself, and that the only factors of 23 are 1 and 23 itself.

Prime number

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

The first seven prime numbers are

2, 3, 5, 7, 11, 13, and 17

The number 1 is not considered to be a prime number, and the number 2 is the first and only even prime number.
Many numbers have factors other than themselves and 1. For example, the factors of 28 are 1, 2, 4, 7, 14, and 28 (since each of these whole numbers and only these whole numbers divide into 28 without a remainder).

Composite numbers

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

Some composite numbers are 4, 6, 8, 10, 12, and 15.

The fundamental principle of arithmetic

Prime numbers are very important in the study of mathematics. We will use them soon in our study of fractions. We will now, however, be introduced to an important mathematical principle.

The fundamental principle of arithmetic

Except for the order of the factors, every whole number, other than 1, can be factored in one and only one way as a product of prime numbers.

Prime factorization

When a number is factored so that all its factors are prime numbers, the factorization is called the prime factorization of the number.

Sample set a

Find the prime factorization of 10.

10 = 2 · 5

Both 2 and 5 are prime numbers. Thus, 2 · 5 is the prime factorization of 10.

Find the prime factorization of 60.

60 = 2 · 30 30 is not prime . 30 = 2 · 15 = 2 · 2 · 15 15  is not prime . 15 = 3 · 5 = 2 · 2 · 3 · 5 We'll use exponents .  2 · 2 = 2 2 = 2 2 · 3 · 5

The numbers 2, 3, and 5 are all primes. Thus, 2 2 · 3 · 5 is the prime factorization of 60.

Find the prime factorization of 11.

11 is a prime number. Prime factorization applies only to composite numbers.

The prime factorization of a whole number

The following method provides a way of finding the prime factorization of a whole number. The examples that follow will use the method and make it more clear.

  1. Divide the number repeatedly by the smallest prime number that will divide into the number without a remainder.
  2. When the prime number used in step 1 no longer divides into the given number without a remainder, repeat the process with the next largest prime number.
  3. Continue this process until the quotient is 1.
  4. The prime factorization of the given number is the product of all these prime divisors.

Sample set b

Find the prime factorization of 60.

Since 60 is an even number, it is divisible by 2. We will repeatedly divide by 2 until we no longer can (when we start getting a remainder). We shall divide in the following way.

The prime factorization of sixty. See the longdesc for a full description.    30 is divisible by 2 again . 15 is not divisible by 2, but is divisible by 3, the next largest prime . 5 is not divisible by 3, but is divisible by 5, the next largest prime . The quotient is 1 so we stop the division process .

The prime factorization of 60 is the product of all these divisors.

60 = 2 · 2 · 3 · 5 We will use exponents when possible . 60 = 2 2 · 3 · 5

Find the prime factorization of 441.

Since 441 is an odd number, it is not divisible by 2. We’ll try 3, the next largest prime.

The prime factorization of four hundred forty-one. See the longdesc for a full description.    147 is divisible by 3 . 49 is not divisible by 3 nor by 5 ,  but by 7 . 7 is divisible by 7 . The quotient is 1 so we stop the division process .

The prime factorization of 441 is the product of all the divisors.

441 = 3 · 3 · 7 · 7 We will use exponents when possible . 441 = 3 2 · 7 2

Exercises

For the following problems, determine which whole numbers are prime and which are composite.

23

prime

25

27

composite

2

3

prime

5

7

prime

9

11

prime

34

55

composite

63

1044

composite

339

209

composite

For the following problems, find the prime factorization of each whole number. Use exponents on repeated factors.

26

38

2 · 19

54

62

2 · 31

56

176

2 4 · 11

480

819

3 2 · 7 · 13

2025

148,225

5 2 · 7 2 · 11 2

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Basic mathematics review. OpenStax CNX. Jun 06, 2012 Download for free at http://cnx.org/content/col11427/1.2
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