# 3.3 Prime factorization of natural numbers

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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

$\left(\text{factor}\right)\cdot \text{}\left(\text{factor}\right)=\text{product}$

## 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 $\text{27}÷3=9$ , or $3\cdot 9=\text{27}$ .

7 is a factor of 56, since $\text{56}÷7=8$ , or $7\cdot 8=\text{56}$ .

4 is not a factor of 10, since $\text{10}÷4=2R2$ . (There is a remainder.)

## 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.

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.

## Practice set a

Find all the factors of each of the following numbers.

6

1, 2, 3, 6

12

1, 2, 3, 4, 6, 12

18

1, 2, 3, 6, 9, 18

5

1, 5

10

1, 2, 5, 10

33

1, 3, 11, 33

19

1, 19

## 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.

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