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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:

  1. Divide the number by the smallest possible prime number.
  2. Divide the result by the smallest possible prime number.
  3. 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.

Vocabulary

Exponent
An exponent tells how many times to multiply a number or term (the "base" ). The exponent is called a power. In 63, 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:
x3, x4
(ab)2, (ab)y
2x3, x2
NOT Like Bases:
y3, x3
(xy)2, y2
(2x)3, x3

 

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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.