Prime Factorisation Method

WHAT IS PRIME FACTORISATION METHOD?

Prime factorisation is the method of breaking down a number into a product of its prime factors. What are prime numbers? They are the numbers that have only two factors – the number itself and 1. A prime number cannot be broken down any further. Thus, prime factorisation is simply representing a number as the product of its prime factors. For example, consider the number 18. 18 can be written as 9 x 2 = 18 but 9 is not a prime number so we can break 9 further into 3 x 3 where 3 is a prime number. 

So, prime factorisation of 18 = 3 x 3 x 2 or 18 = 3² x 2

Let us take another example: Consider the number 50

50 can be written as 5 x 10. Here, 5 is a prime number and cannot be broken down any further but 10 is not a prime number. 10 can be written as 2 x 5. 

Therefore, prime factorisation of 50 = 2 x 5 x 5 or 50 = 2 x 5²

The word prime factorisation consists of prime and factor. We already know the meaning of prime, let us understand what factors are. Factors are simply the numbers that can divide the original number and cannot be further divided into any more factors.

PRIME FACTORISATION METHODS

There are many methods to find the prime factors of a number but the commonly used two methods are as follows:

1. Division method
2. Factor tree method

DIVISION METHOD

The division method is as the word suggests, we divide the numbers with the least possible prime number and keep dividing it till the quotient is 1. Let us see how it is done in the given steps:

1. Start by dividing the number by the least possible prime number. The least prime number should divide it completely.
2. Repeat step 1 and keep dividing the resultant number with the least possible prime number and keep doing that until the quotient is 1.

Let us have a look at an example to better understand how it is done:

Consider the number 160

The prime factorisation of 160 = 2 x 2 x 2 x 2 x 2 x 5 or 160 =  2⁵ x 5

FACTOR TREE METHOD

As the name suggests, we have a tree-like structure here where we consider the original number as the root of the tree.

We follow the given steps:

1. Take the original number as the root of the tree.
2. Break down the number into factors and write it as the branches of the tree.
3. Factorize the composite numbers to break them into smaller prime numbers.
4. Keep factoring until all is left is prime numbers

Look at the following image to understand the factor tree method. Consider the original number as 80

As we can see, the numbers in yellow boxes are prime numbers. 

80 = 2 x 2 x 2 x 2 x 5 or 80 = 2⁴ x 5

Illustrations

1. What is the prime factorisation of 1260?
Ans: Let us solve using the division method. We will get the answer as –
Therefore, prime factorisation of 1260 = 2 x 2 x 3 x 3 x 5 x 7

2. What is the prime factorisation of 540?
Ans: We can find the solution as follow:
Therefore, prime factorisation of 540 = 2 x 2 x 3 x 3 x 3 x 5

3. What is the prime factorisation of 19?
Ans: Since 19 is a prime number and cannot be broken down any further so prime factorisation of 19 = 1 x 19.