Prime Factorization

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.

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 = 2² × 3
• Prime factorization of 18 is 2 × 3 × 3 = 2 × 3²
• Prime factorization of 24 is 2 × 2 × 2 × 3 = 2³ × 3
• Prime factorization of 20 is 2 × 2 × 5 = 2² × 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 step-by-step 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 2² 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.

∴ The Prime Factors of 1240 will be 2³ × 5 × 31.

Q.2: Find the prime factors of 544.

Solution:

Therefore, the prime factors of 544 are 2⁵ x 17.

Prime Factorization Worksheet (Questions)

1. What is the prime factorization of 48?
2. Write the prime factors of 2664 without using exponents.
3. Is 40 = 20 × 2 an example of prime factorization process? Justify.
4. Write 6393 as a product of prime factors.

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