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 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. For example, Prime factorization of;

  • 12 is 2 x 2 x 3 = 22 x 3
  • 18 is 2 x 3 x 3 = 2 x 32
  • 24 is 2 x 2 x 2 x 3 = 23 x 3
  • 20 is 2 x 2 x 5 = 22 x 5

What is Prime Factorization?

Prime factorization is defined as a way of finding the prime factors of a number i.e. the prime numbers which can be multiplied together to get the original number. For example, the prime factors of 126 will be 2, 3 and 7 as 2 × 3 × 7 = 126 and 2, 3, 7 are primes.

The prime numbers when multiplied by any natural numbers or whole numbers ( but not 0), gives composite numbers. So basically prime factorisation is performed on the composite numbers to factorize them and find the prime factors. This method is also used in 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. 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.

Factor Tree

In factor tree, the factors of a number are found and then those numbers are further factorised 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.

Factor tree

In the above figure, we can number 60 is first factorised into two numbers i.e. 6 and 10. Again, 6 and 10 is factorised 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;

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 the cases, a tree structure is formed.

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Prime Factorization Method Steps

The steps to calculate the prime factors of a number is similar to the process of finding the factors of a large number. First, the number has to be divided exactly with the least possible prime number which can be 2, 3, 5, etc. This process has to be continued until the end result is 1. Now, the prime factors of that number will be the prime numbers that have divided the number in this process.

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 ÷ 2 = 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

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 22 × 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 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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