# Prime factor decomposition

**Prime factors**

A **prime factor** is a factor which is also a prime number.

For example, the prime factors of 70 are:

The **prime factors** of 70 are 2, 5, and 7.

**Products of prime factors**

70 = 2 × 5 × 7

56 = 2 × 2 × 2 × 7 This can also be 56 = 23 × 7

99 = 3 × 3 × 11 This can also be 99 = 32 × 11

Therefore every whole number more than 1 is either:

- A prime number, or
- Can be written as a product of two or more prime numbers.

This concept is also known as the **Fundamental Theorem of Arithmetic. **Also, numbers which are not prime are often called composite numbers because they are made up of products of prime numbers.

**The prime factor decomposition**

**P****rime factor decomposition **or** prime factor form **is the number which is written as a product of prime factors.

For example the prime factor decomposition of 100 is:

100 = 2 x 2 x 5 x 5

= 2² x 5²

There are two methods of finding the prime factor decomposition of a number. These methods are called Factor trees and Dividing by prime numbers.

**Factor trees**

Factor trees can be used to find the prime factor decomposition of a number:

36 = 2 x 2 x 3 x 3

= 2² x 3²

- Start by writing 36 as the product so at the top
- Think of two numbers which multiply to 36
- Then find two numbers which multiply to the first two numbers which multiply to 36
- As 2 & 3 are prime numbers is at the lowest level

Note: this factor tree for 36 can be completed in other ways, but remember that the primary number is always at the lowest level branch.

**Dividing by prime numbers**

Another method of finding a primary factor decomposition is to repeatedly divide by primary factors and inserting the solutions in a table as shown below:

2 | 96 |

2 | 48 |

2 | 24 |

2 | 12 |

2 | 6 |

3 | 3 |

1 |

96 = 2 x 2 x 2 x 2 x 2 x 3

= 2⁵ x 3

This method requires the following steps:

- To find the prime factor decomposition of 96 start with it at the top of the table
- Find the lowest prime number that divides 96, this is 2. Write this on the left side of 96 then divide by two
- Insert this number below 96 and carry on until the lowest number on the table is the lowest prime number.
- Below the lowest prime number will be 1.

Therefore the prime factor decomposition will be found by multiplying all numbers on the left hand side column of the table.

**Using the prime factor decomposition**

The prime factor decomposition can then be used to deduce that the number is a square number. See example below:

2 | 324 |

2 | 162 |

3 | 81 |

3 | 27 |

3 | 9 |

3 | 3 |

1 |

324 = 2 x 2 x 3 x 3 x 3 x 3

= 2² x 3⁴

This can be written as:

(2 x 3²) x (2 x 3²)

or

(2 x 3²)²

If all the indices in the prime factor decomposition of a number are even, then the number is a square number.

Based on the example above, it is apparent that:

- 324 = (2 × 3 × 3) × (2 × 3 × 3)
- (square root) √324 = 2 × 3² = 2 × 9 = 18.