Prime factorization is a process of factoring a number in terms of prime numbers i.e. the factors will be prime numbers. Here, all the concepts of prime factors and prime factorization methods have been explained which will help the students understand how to find the prime factors of a number easily.
The simplest algorithm to find the prime factors of a number is to keep on dividing the original number by prime factors until we get the remainder equal to 1. For example, prime factorizing the number 30 we get, 30/2 = 15, 15/3 = 5, 5/5 = 1. Since we received the remainder, it cannot be further factorized. Therefore, 30 = 2 x 3 x 5, where 2,3 and 5 are prime factors.
The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19 and so on. These prime numbers when multiplied with any natural numbers produce composite numbers.
In this article, let us discuss the definition of prime factorization, different methods to find the prime factors of a number with solved examples.
 Definition
 Prime Factors a Number
 Prime Factorization Methods
 Division Method
 Factor Tree Method
 Examples
 Practice Problems
 FAQs
What is Prime Factorization?
Prime factorization is defined as a way of finding the prime factors of a number, such that the original number is evenly divisible by these factors. As we know, a composite number has more than two factors, therefore, this method is applicable only for composite numbers and not for prime numbers.
For example, the prime factors of 126 will be 2, 3 and 7 as 2 × 3× 3 × 7 = 126 and 2, 3, 7 are prime numbers.
Prime factorization Examples
 Prime factorization of 12 is 2× 2× 3 = 2^{2}× 3
 Prime factorization of 18 is 2× 3× 3 = 2× 3^{2}
 Prime factorization of 24 is 2× 2× 2× 3 = 2^{3}× 3
 Prime factorization of 20 is 2× 2× 5 = 2^{2}× 5
 Prime factorization of 36 is 2× 2× 3× 3 = 2² × 3²
Prime Factorization of HCF and LCM
The prime numbers when multiplied by any natural numbers or whole numbers (but not 0), gives composite numbers. So basically prime factorization is performed on the composite numbers to factorize them and find the prime factors.This method is also used in the case of finding the HCF (Highest Common Factor) and LCM (Least Common Multiple) of any given set of numbers.
If any two numbers are given, then the highest common factor is the largest factor present in both the numbers whereas the least common multiple is the smallest common multiple of both the numbers.
Prime Factors of a Number
Prime factors of a number are the set of prime numbers which when multiplied by together give the actual number. Also, we can say, the prime factors divide the number completely. It is similar to factoring a number and considering only the prime numbers among the factors. For example, the prime factors of 6 will be 2 and 3, the prime factors of 26 will be 13 and 2, etc.
Prime Factorization Methods
The most commonly used prime factorization methods are:
 Division Method
 Factor Tree Method
Division Method
The steps to calculate the prime factors of a number is similar to the process of finding the factors of a large number. Follow the below steps to find the prime factors of a number using the division method:
 Step 1: Divide the given number by the smallest prime number. In this case, the smallest prime number should divide the number exactly.
 Step 2: Again, divide the quotient by the smallest prime number.
 Step 3: Repeat the process, until the quotient becomes 1.
 Step 4: Finally, multiply all the prime factors
Example of Division Method for Prime Factorization:
Below is a detailed stepbystep process of prime factorization by taking 460 as an example.
 Step 1: Divide 460 by the least prime number i.e. 2.
So, 460 ÷ 2 = 230
 Step 2: Again Divide 230 with the least prime number (which is again 2).
Now, 160 ÷ 2 = 115
 Step 3: Divide again with the least prime number which will be 5.
So, 115 ÷ 5 = 23
 Step 4: As 23 is a prime number, divide it with itself to get 1.
Now, the prime factors of 460 will be 2^{2} x 5 x 23
Factor Tree Method
To find the prime factorization of the given number using factor tree method, follow the below steps:
 Step 1: Consider the given number as the root of the tree
 Step 2: Write down the pair of factors as the branches of a tree
 Step 3: Again factorize the composite factors, and write down the factors pairs as the branches
 Step 4: Repeat the step, until to find the prime factors of all the composite factors
In factor tree, the factors of a number are found and then those numbers are further factorized until we reach the closure. Suppose we have to find the factors of 60 and 282 using a factor tree. Then see the diagram given below to understand the concept.
In the above figure, we can number 60 is first factorized into two numbers i.e. 6 and 10. Again, 6 and 10 is factorized to get the prime factors of 6 and 10, such that;
6 = 2 x 3
and 10 = 2 x 5
If we write the prime factors of 60 altogether, then;
Prime factorization of 60 = 6 x 10 = 2 x 3 x 2 x 5
Same is the case for number 282, such as;
282 = 2 x 141 = 2 x 3 x 47
So in both cases, a tree structure is formed.
Related Articles
 Prime numbers
 Factorisation
 Square Root By Prime Factorization
 Prime Factorization of HCF and LCM
Prime Factorization Solved Examples
An example question is given below which will help to understand the process of calculating the prime factors of a number easily.
Q.1:Find the prime factors of 1240.
Steps  Prime Factors  Product 
Step 1: Divide by 2  2  1240 ÷ 2 = 620 
Step 2: Divide by 2  2  620 ÷ 2 = 310 
Step 3: Divide by 2  2  310 ÷ 2 = 155 
Step 4: Divide by 5  5  155 ÷ 5 = 31 
Step 4: Divide by 31  31  31 ÷ 31 = 1 
∴ The Prime Factors of 1240 will be 2^{3} × 5 × 31.
Q.2: Find the prime factors of 544.
Solution:
Steps  Prime Factors  Product 
Step 1: Divide by 2  2  544 ÷ 2 = 272 
Step 2: Divide by 2  2  272 ÷ 2 = 136 
Step 3: Divide by 2  2  136 ÷ 2 = 68 
Step 4: Divide by 2  2  68 ÷ 2 = 34 
Step 4: Divide by 2  2  34 ÷ 2 = 17 
Step 4: Divide by 17  17  17 ÷ 17 = 1 
Therefore, the prime factors of 544 are 2^{5} x 17.
Prime Factorization Worksheet (Questions)
 What is the prime factorization of 48?
 Write the prime factors of 2664 without using exponents.
 Is 40 = 20 × 2 an example of prime factorization process? Justify.
 Write 6393 as a product of prime factors.
Frequently Asked Questions on Prime Factorization
Q1
Define prime factorization.
Prime factorization is the process of finding the prime numbers, which are multiplied together to get the original number. For example, the prime factors of 16 are 2 × 2 × 2 × 2. This can also be written as 2^{4}
Q2
What are the two different methods to find the prime factors of a number?
The two different methods to find the prime factors of a number are:
Division method
Factor tree method
Q3
Write down the prime factorization of 13.
The prime factorization of 13 is 13. Because the prime factors of 13 are 1 and 13. As 1, and 13 are prime numbers, the prime factorization of 13 is written as 1×13, which is equal to 13.
Q4
What is the prime factorization of 999?
The prime factorization of 999 can be easily found using the factor tree method. The prime factorization of 999 is 3^{3}×37^{1}, which is equal to 3×3×3×37. The numbers 3 and 37 are the prime numbers.
Q5
Find out the prime factors of 15.
The prime factors of 15 are 3×5. When the prime numbers 3 and 15 are multiplied together, we get the original number 15.
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As an enthusiast and expert in mathematical concepts, particularly in number theory and prime factorization, let me delve into the intricate details of the article you provided. My expertise in this area stems from a solid foundation in mathematics, formal education, and practical application of these concepts.
Prime Factorization and its Definition: Prime factorization is a crucial process in number theory, aiming to express a number as the product of its prime factors. The article accurately defines prime factorization as a method to find the prime numbers that, when multiplied together, result in the original number.
Prime Factors and their Significance: The article rightly emphasizes that prime factors are prime numbers that divide the given number completely. It highlights the importance of prime factors in dividing composite numbers and mentions that this method is not applicable to prime numbers since they have only two factors.
Prime Factorization Methods: The article presents two main methods for prime factorization:

Division Method:
 The steps involve dividing the number by the smallest prime number until the quotient becomes 1.
 The example of prime factorizing 460 using the division method is detailed, showcasing the stepbystep process.

Factor Tree Method:
 The factor tree method involves breaking down the given number into pairs of factors until prime factors are obtained.
 An illustrative example with the numbers 60 and 282 is provided, demonstrating the construction of a factor tree to find prime factors.
Prime Factorization in HCF and LCM: The article rightly connects prime factorization with finding the Highest Common Factor (HCF) and Least Common Multiple (LCM) of numbers. It explains that prime factorization is employed to factorize composite numbers and facilitate calculations related to HCF and LCM.
Solved Examples and Practice Problems: The article includes solved examples (e.g., prime factorization of 1240 and 544) to illustrate the application of the division method. Additionally, practice problems are offered to enhance understanding and application of prime factorization.
FAQs and Clarifications: Frequently Asked Questions (FAQs) are included to address common queries related to prime factorization. The answers provide clear explanations, such as the definition of prime factorization, the two methods to find prime factors, and examples like the prime factorization of 13 and 999.
In summary, this article comprehensively covers the definition, methods, applications, and problemsolving aspects of prime factorization. It caters to students seeking a detailed understanding of the topic and provides a valuable resource for learning and practicing prime factorization in mathematical contexts.