1.2 Prime factorization

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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.

$10=2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5$

Both 2 and 5 are prime numbers. Thus, $2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5$ is the prime factorization of 10.

Find the prime factorization of 60.

$\begin{array}{lllll}60\hfill & =\hfill & 2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}30\hfill & \hfill & \text{30\hspace{0.17em}is\hspace{0.17em}not\hspace{0.17em}prime}.\text{\hspace{0.17em}}30=2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}15\hfill \\ \hfill & =\hfill & 2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}15\hfill & \hfill & 15\text{\hspace{0.17em}is\hspace{0.17em}not\hspace{0.17em}prime}.\text{\hspace{0.17em}}15=3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\hfill \\ \hfill & =\hfill & 2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\hfill & \hfill & \text{We'll\hspace{0.17em}use\hspace{0.17em}exponents}\text{.\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2={2}^{2}\text{\hspace{0.17em}}\hfill \\ \hfill & =\hfill & {2}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\hfill & \hfill & \hfill \end{array}$

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.

$\begin{array}{l}\text{30\hspace{0.17em}is\hspace{0.17em}divisible\hspace{0.17em}by\hspace{0.17em}2\hspace{0.17em}again}\text{.}\hfill \\ \text{15\hspace{0.17em}is\hspace{0.17em}not\hspace{0.17em}divisible\hspace{0.17em}by\hspace{0.17em}2,\hspace{0.17em}but\hspace{0.17em}is\hspace{0.17em}divisible\hspace{0.17em}by\hspace{0.17em}3,\hspace{0.17em}the\hspace{0.17em}next\hspace{0.17em}largest\hspace{0.17em}prime}\text{.}\hfill \\ \text{5\hspace{0.17em}is\hspace{0.17em}not\hspace{0.17em}divisible\hspace{0.17em}by\hspace{0.17em}3,\hspace{0.17em}but\hspace{0.17em}is\hspace{0.17em}divisible\hspace{0.17em}by\hspace{0.17em}5,\hspace{0.17em}the\hspace{0.17em}next\hspace{0.17em}largest\hspace{0.17em}prime}\text{.}\hfill \\ \text{The\hspace{0.17em}quotient\hspace{0.17em}is\hspace{0.17em}1\hspace{0.17em}so\hspace{0.17em}we\hspace{0.17em}stop\hspace{0.17em}the\hspace{0.17em}division\hspace{0.17em}process}\text{.}\hfill \end{array}$

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

$\begin{array}{lllll}60\hfill & =\hfill & 2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\hfill & \hfill & \text{We\hspace{0.17em}will\hspace{0.17em}use\hspace{0.17em}exponents\hspace{0.17em}when\hspace{0.17em}possible}.\hfill \\ 60\hfill & =\hfill & {2}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5\hfill & \hfill & \hfill \end{array}$

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.

$\begin{array}{l}\text{147\hspace{0.17em}is\hspace{0.17em}divisible\hspace{0.17em}by\hspace{0.17em}3}.\hfill \\ \text{49\hspace{0.17em}is\hspace{0.17em}not\hspace{0.17em}divisible\hspace{0.17em}by\hspace{0.17em}3\hspace{0.17em}nor\hspace{0.17em}by\hspace{0.17em}5},\text{\hspace{0.17em}but\hspace{0.17em}by\hspace{0.17em}7}.\hfill \\ \text{7\hspace{0.17em}is\hspace{0.17em}divisible\hspace{0.17em}by\hspace{0.17em}7}\text{.}\hfill \\ \text{The\hspace{0.17em}quotient\hspace{0.17em}is\hspace{0.17em}1\hspace{0.17em}so\hspace{0.17em}we\hspace{0.17em}stop\hspace{0.17em}the\hspace{0.17em}division\hspace{0.17em}process}.\hfill \end{array}$

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

$\begin{array}{lllll}441\hfill & =\hfill & 3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7\hfill & \hfill & \text{We\hspace{0.17em}will\hspace{0.17em}use\hspace{0.17em}exponents\hspace{0.17em}when\hspace{0.17em}possible}.\hfill \\ 441\hfill & =\hfill & {3}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{7}^{2}\hfill & \hfill & \hfill \end{array}$

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

$2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}19$

54

62

$2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}31$

56

176

${2}^{4}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}11$

480

819

${3}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}7\text{\hspace{0.17em}}·\text{\hspace{0.17em}}13$

2025

148,225

${5}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{7}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{11}^{2}$

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