Prime factorization: Definition, Methods, Applications

In this post, I am going to show you what Prime Factorization is and how to find the Prime Factorization of a number.

Let’s go!

Prime Factorization

First of all, you need to know what a prime number is.

A prime number is a number that has only two factors, 1 and the number itself. 

For example, the only two factors of 2 are 2 and 1, and of 11 are 11 and 1.

Notice that 1 isn’t considered prime since 1 has only one factor (itself).

The number 2 is the smallest prime and the only even prime.

Here is a list of the first 30 prime numbers:

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

A composite number is a natural number which has more than two factors.

Ex: 8 has more than two factors (1, 2, 4, 8)

What is Prime Factorization?

The process of writing a number into prime factors is Prime Factorization.

For example, 30 = 2 x 3 x 5; 729 = 2³ x 3² x 11

How to find Prime Factorization

We have 2 methods: Repeated Division and Factor Tree

#Method 1. Repeated Division

To find the prime factors of a composite number, we divide the number by the prime numbers which are its factors, starting with the smallest.

Example 1: Find the prime factorization of 60.

It is best to start working from the smallest prime number, which is 2, so let’s check:

60 ÷ 2 = 30

Yes, 60 divided exactly by 2. We have taken the first step!

But 30 is not a prime number, so we need to go further. Let’s try 2 again:

30 ÷ 2 = 15 (15 is not a prime number, we need to try factoring 15, so we divide 15 by its prime factor 3)

15 ÷ 3 = 5. And 5 is a prime number, so we have the answer:

60 = 2 × 2 × 3 × 5 

Note: 60 = 2 × 2 × 3 × 5 can also be written using exponents as 60 = 2² × 3 × 5

Example 2: What is the prime factorization of 135?

Let’s start with the smallest prime number which is a factor of 135.

135 ÷ 3 = 45

45 ÷ 3 = 15

15 ÷ 3 = 5

And 5 is a prime number, so we have the answer: 135 = 3 × 3 × 3 × 5  = 3³ × 5

You can also write like this:

I have shown you how to do the factorization by repeated division, but sometimes it’s easier to break a number down into any factors you can.

Example: What are the prime factors of 70?

We break 70 into 7 x 10

The prime factors of 10 are 2 and 5. And 7 is a prime number.

Then 70 = 2 x 5 x 7

So the prime factors of 70 are 2, 5, 7.

#Method 2: Factor tree

We have another method which calls ‘Factor tree’.

This method can help: find any factors of the number, then the factors of those numbers, …

For example, 72 = 8 x 9 so we write 8 and 9 below 72.

We have found the prime factors of 72 when we can’t factor anymore.

More:

Prime Factorization Calculator

Applications of Prime factorization

#1. Find HCF

We can use Prime factorization to find the highest common factor (HCF) of two or more natural numbers.

A number which is a factor of two or more numbers is called a common factor of these numbers.

For example, 5 is a common factor or 20 and 25 since 5 is a factor of both of these numbers.

The HCF (highest common factor) of 2 or more number is the largest factor that is common to all the number.

Example: Find the highest common factor of 18 and 30.

Solution:

Firstly, we can use the method of finding prime factors of 18 and 30 (Prime factorization by repeated division or tree factor).

Therefore, we have: 18 = 2 x 3 x 3 and 30 = 2 x 3 x 5

So, 2, 3, and 6 are common factors of 18 and 30.

The HCF of 18 and 30 is 2 x 3 = 6.

#2. Find LCM

Prime factorization can be used to find the LCM (the lowest common multiple) of two or more numbers.

The multiples of any whole number have that number as a factor. They are obtained by multiplying it by 1, then 2, then 3, and so on.

For example, the multiples of 3 are: 3 (=3 x 1), 6 (=3 x 2), 9 (=3 x 3), 12 (=3 x 4), ….

The LCM (the lowest common multiple) of two or more numbers is the smallest mutiple that is common to all the numbers.

Example: Find the lowest common multiple of 252 and 616.

Solution:

We need express each number as a product of its prime factors first. (Prime factorization)

252 = 2² x 3² x 11

616 = 2³ x 7 x 11

The LCM = 2³ x 3² x 7 x 11 = 5544.

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