## Factoring

## Overview: Factoring Numbers

When you factor a number, you break it apart to find the integers that can be multiplied together to produce the original number.

### Factor the number 30

30 = (1)(30)

30 = (2)(15)

30 = (3)(10)

30 = (5)(6)

1,2,3,5,6,10,15,30 are all factors of 30.

### Factors can be composite numbers or prime numbers.

A composite number is any number that has more than two factors.

In the example above, 15, 10, and 6 are composite numbers. They can be broken down further.

15 = (3)(5)

10 = (2)(5)

6 = (2)(3)

A prime number is a positive number greater than 1 that has exactly two divisors: 1 and itself.

### The first 30 primes:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29,

31, 37, 41, 43, 47, 53, 59, 61, 67,

71, 73, 79, 83, 89, 97, 101, 103,

107, 109, 113

(0 and 1 are not considered prime or composite.)

### Prime Factorization

To find the prime factors of a number:

- Divide the number by the smallest possible prime number.
- Divide the result by the smallest possible prime number.
- Continue until the result is also a prime number.

### Find the prime factorization of 30

30 ÷ 2 = 15

15 ÷ 3 = 5

2,3,5 are the prime factors of 30.

Check your answer: (2)(3)(5) = 30

When you multiply all of the prime factors of a number,

the product (result) is the original number.

### Find the prime factorization of 168

168 ÷ 2 = 84

84 ÷ 2 = 42

42 ÷ 2 = 21

21 ÷ 3 = 7

2,2,2,3,7 are the prime factors of 168.

Check your answer: (2)(2)(2)(3)(7) = 168

Factoring is useful in algebra for simplifying expressions and solving equations.

- Exponent
- An exponent tells how many times to multiply a number or term (the "base" ). The exponent is called a power. In 6
^{3}, we read "Six to the third power," and 6 is multiplied 3 times: (6)(6)(6). - Like Bases
- The base is the number or term that is raised to a power. Like bases have the same number or term raised to a power. The power (exponent) may be different.
**Like Bases:**

x^{3}, x^{4}

(ab)^{2}, (ab)^{y}

2x^{3}, x^{2}**NOT Like Bases**:

y^{3}, x^{3}

(xy)^{2}, y^{2}

(2x)^{3}, x^{3}

## Vocabulary

This material is based upon work supported by the National Science Foundation (NSF) under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.