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# Prime Factorization – Explanation & Examples

**Prime factorization** is a method of finding all the prime numbers that multiply to form a number. Factors are multiplied to get a number, while prime factors are the numbers that can only be divided by 1 or themselves.

## How to find Prime Factorization?

There are two methods of finding prime factors of a number. These are repeated division and factor tree.

### Repeated division

A number is reduced by dividing it severally with prime numbers. Prime factors of number 36 are found by repeated division as shown:

The prime factors of number 36 are, therefore, 2 and 3. This can be written as 2 × 2 × 3 × 3. It is advisable to start dividing a number by the smallest prime number and proceed to bigger factors.

*Example 1*

*What are the prime factors of 16?*

__Solution__

The best way to solve this problem is by identifying the smallest prime factor of the number, which is 2.

Divide number by 16;

16 ÷ 2 = 8

Because 8 is not a prime number, proceed by dividing again by the smallest factor;

8 ÷ 2 = 4

4 ÷ 2 = 2

We have the prime factors of 16 highlighted in yellow, and they include: 2 x 2 x 2 x 2.

which can be written as an exponent:

16 = __2 ^{2}__

* *

*Example 2*

*Find the prime factors of 12.*

__Solution__

Divide 12 by 2;

12 ÷ 2 = 6

6 is not prime, proceed;

6 ÷ 2 = 3.

Therefore, 12 = 2 x 2 x 3

12 = 2 ^{2 }× 3

It is noted that, all prime factors of a number are prime.

*Example 3*

*Factorize 147.*

__Solution__

Start by dividing 147 by the smallest prime number.

147 ÷ 2 = 73.5

Our answer isn’t an integer, try the next prime number 3.

147 ÷ 3 = 49

Yes, 3 worked, now proceed to the next prime that can divide 49.

49 ÷ 7 = 7

Therefore, 147 = __3 x 7 x 7,__

=__3 x 7 ^{2}.__

* *

*Example 4*

*What is the prime factorization of 19?*

19 =__ 19__

__Solution__

Another method on how to perform factorization is to break a number down into two integers. Now find the prime factors of the integers. This technique is useful when dealing with bigger numbers.

* *

*Example 5*

*Find the prime factors of 210.*

__Solution__

Break down 210 into:

210 = 21 x 10

Now calculate the factors of 21 and 10

21 ÷ 3 = 7

10 ÷ 2 = 5

Combine the factors: 210 = __2 x 3 x 5 x 7__

### Factor tree

Factor tree involves finding the prime factors of a number by drawing tree- like programs. Factor tree is the best tool of doing prime factorization. The prime factors of 36 are obtained by factor tree as shown below: