Session 4: Prime Factorisation – Definition, Methods, Steps and Examples

What is Prime Factorisation?

When a natural number is factorised, it is broken down into two factors that are either prime numbers or composite numbers. Prime numbers are numbers whose only factors are 1 and the number itself. Composite numbers are numbers that have more than two factors. The number of factors that a number has determines whether the number is a prime number or a composite number. If composite, the factors can be split up further. If we keep on doing this, ultimately, we arrive at a stage when all the factors are prime factors. This process of breaking down a number into a product of all prime numbers is called prime factorisation. 

Methods of Prime Factorisation

There are various methods to do the prime factorisation of a number. The most common methods are:

• Prime factorisation using the factor tree method
• Prime factorisation using the division method

Prime Factorisation using Factor Tree Method

A factor tree is a diagram used to determine the prime factors of a natural number greater than one. In the factor tree method, we start by writing the number and then write down the corresponding pair of factors below the number. We connect the two factors to the number with small line segments as the branches of the factor tree. If a factor is prime, we circle it and do not factor that “branch” any further. If a factor is composite, we repeat this process, writing down the pair of factors as the new branches of the tree.
We continue until we get the prime factors of all the composite factors. The circled primes give us the prime factorisation when the factor tree is complete.
For example, let’s do the prime factorisation of 198 by the factor tree method. 
We can start with any factor pair, such as 2 and 99. We write 2 and 99 below 198 with branches connecting them.

The factor 2 is prime, so we circle it. Factor 99 is composite, so we need to factorise it further. 

The factor 3 is prime, so we circle it. 
Factor 33 is composite, so we factorise it. The resulting factors 3 and 11 are prime, so we circle them.

The prime factorisation is the product of the prime factors that are circled. 
198=2×3×3×11

Note: By convention, the prime factors are listed from least to greatest.

Steps to Prime Factorise a Number using the Factor Tree Method

To find the prime factorisation of a number using the factor tree method, we use the following steps:
Step 1: Find any factor pair of the given number and write these numbers as the branches of the tree.
Step 2: If a factor is prime, that branch is complete. Circle the prime number.
Step 3: If a factor is composite, factorise the number and write down the pair of factors as the next branches of the tree.
Step 4: Repeat step 3 until we get the prime factors of all the composite factors.
Step 5: Write the given number as the product of all the circled primes.

Fundamental Theorem of Arithmetic or Prime Factorisation Property

1) Factorise 36. 
Given below are several ways to factorise 36. 

All the composite numbers can be factorised further. The ringed numbers cannot be factorised further; hence these factors are called prime factors. 

 

 

In the above tree diagrams, we observe that though prime factors appear in a different order in each case, all combinations give the same set of prime factors. 
Thus, the prime factorisation of 36 is unique. This is not only true for 36, but it holds for every composite number. 
Thus, every natural number can be expressed in the form of the product of the prime factors. This property is known as the Prime Factorisation Property or Fundamental Theorem of Arithmetic. 

Prime Factorisation using Division Method

The division method is another way to find the prime factors of a composite number. It leads to the same result as the factor tree method. For large numbers, the factor tree method is lengthy and time-consuming. To overcome this drawback, the division method is used to find the prime factors.  
In the division method, we divide the given number by its smallest prime factor. Again, divide the quotient obtained by its smallest prime factor. We repeat this process until the quotient becomes 1.
Now, write the given number as the product of all the primes that are the divisors of the division.
For example, let’s factorise 36 by the division method. 
We start by dividing 36 by 2, the smallest prime factor of 36. 

We continue dividing by the same prime until it no longer divides evenly. 

Then, we divide by the next prime, so we divide 9 by 3. 
We continue dividing until the quotient is 1. 

Since the quotient is 1, we stop here. The prime factorisation is the product of all the divisors on the left side

36=2×2×3×3

Steps to Prime Factorise a Number using the Division Method

To find the prime factorisation of a number using the division method; we use the following steps:
Step 1: Divide the given number by its smallest prime factor.
Step 2: Divide the quotient obtained in step 1 by its smallest prime factor.
Step 3: Continue until the quotient is a 1.
Step 4: Write the given number as the product of all the primes that are the divisors of the division.

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Prime Factorisation – Examples

Example 1

Using the factor tree method, express each of the following numbers as the product of its prime factors. 
i. 6006
ii. 2744
i. Prime factorisation of 6006: Using the factor tree, we have 

6006=2×3×7×11×13

ii. Prime factorisation of 2744: Using the factor tree, we have

2744=2×2×2×7×7×7

Example 2 

Using the division method, express each of the following numbers as the product of its prime factors. 
i. 1225
ii. 2520
i. Using the division method, we have

∴1225=5×5×7×7

ii. Using the division method, we have

∴2520=2×2×2×3×3×5×7

Example 3

Write the largest 4-digit number and express it as a product of primes using the factor tree and division method. 
The largest 4-digit number is 9999. 
Prime factorisation of 9999 using the factor tree method. 

∴9999=3×3×11×101
Prime factorisation using division method

Example 4

Find the smallest number having six different prime factors. 
The number having six distinct smallest prime numbers as its factors will be the required number.
Hence, the smallest number having six different prime factors=2×3×5×7×11×13=30030

Remember this!

• A factor is a number that divides another number completely without any remainder. 
• The numbers greater than 1 and only have two factors: 1 and the prime number itself are called prime numbers. 
• The numbers that have more than two factors are called composite numbers. 
• 1 is neither prime nor composite. 
• 2 is the smallest prime number and is the only even prime number.