Methods of Prime Factorization

In prime factorization, we factorize the numbers into prime numbers, called prime factors.

Expressing a number as the product of its factors is called factorization.

For example, 48 = 2 × 24

Here, 48 in factorised.

There are many other forms of factorization of 48.

Such as 48 = 2 × 2 × 12

                 = 2 × 2 × 2 × 2 × 3

48 = 2 × 2 × 2 × 2 × 3, we see that each factor of 48 is a prime number.

It is called prime factorization.

When a number is expressed as a product of prime numbers, we call it prime factorization.

or

A factorization is which every factor is prime, is called the prime factorization of the number.


Prime factorization of 30 and 48 are given below:

30 = 2 × 3 × 5

48 = 2 × 2 × 2 × 2 × 3

To make the prime factorization of a number, we divide the given number by the prime number 2, 3, 5, 7, 11, etc., in this order repeatedly so long as the quotient is divisible by that number.



There are two methods of prime factorization:

Factorization means writing a given number as the product of two or more factors.

Prime factorization of a number is a way of showing a number as the product of prime numbers.

There are 2 methods to find the prime factors.

1. Division Method

2. Factor Tree Method


1. Prime Factorization by Division Method:

Observe the following steps.

I: First we divide the number by the smallest prime number which divides the number exactly.

II: We divide the quotient again by the smallest or the next smallest prime number if it is not exactly divisible by the smallest prime number. We repeat the process again and again till the quotient becomes 1. Remember, we use only prime numbers to divide.

III: We multiply all the prime factors. Remember, the product is the number itself.


Let us consider a few examples using division method.

1. Find the prime factors of 15.

First Step: 2 is the smallest prime number. But it cannot divide 15 exactly. So, consider 3.

Second Step: Now, 5 cannot be divide by 3. Consider the next smallest prime number 5.

The prime factors of 15 are 3 × 5.


2. Find the prime factors of 18.

First Step: Consider 2, the smallest prime number.

Second Step: As 9 cannot be divide by 2. Consider the next smallest prime 3. Repeat the process till quotient becomes 1.

The prime factors of 18 are 2 × 3 × 3.


2. Prime Factorization by Factor Tree Method

Observe the following steps.

Suppose, we have to find the prime factors of 16

1. We consider the number 16 as the root of the tree.

2. We write a pair of factors as the branches of the tree i.e., 2 × 8 = 16

3. We further factorize the composite factor 8 as 4 and 2, and again the composite factors 4 as 2 and 2.


We repeat the process again till we get the prime factors of all the composite factors.

2 ×     8                   = 16

2 ×     4  ×  2           = 16

2 ×     2  ×  2 ×  2    = 16

Methods of Prime Factorization






The prime factors of 16 = 2 × 2 × 2 × 2.

We can express the factor tree to find the prime factors of 16 in another way also.

      4           ×           4

2    ×     2    ×     2    ×    2

Method of Prime Factorization





The prime factors of 16 = 2 × 2 × 2 × 2.


Now, we will use both the methods (Factor Tree Method and Division Method) to the prime factorization of a number:

1. The prime factorization of 36

Factor Tree Method

Factor Tree Method

Continue factorizing until only prime number remains.

36 = 2 × 2 × 3 × 3


Division Method

Division Method Factorization

               Steps


          36 ÷ 2 = 18


          18 ÷ 2 = 9


            9 ÷ 3 = 3

Continue diving unit the quotient is a prime.


2. The prime factorization of 64.

Factor Tree Method

Factor Tree Method 64

Continue factorizing until only prime number remains.

64 = 2 × 2 × 2 × 2 × 2 × 2


Division Method 

Division Method Factorization 64

             Steps

          64 ÷ 2 = 32

          32 ÷ 2 = 16

          16 ÷ 2 = 8

            8 ÷ 2 = 4

            4 ÷ 2 = 2

Continue diving unit the quotient is a prime.


More Solved Examples on Methods of Prime Factorization:

3. Express 240 as the product of prime numbers.

Division Method 

Product of Prime Factors

Hence, 240 = 2 × 2 × 2 × 2 × 3 × 5


This process can be shown diagramatically, known as factor tree.

Factor Tree Method:

Factor Tree Method


Therefore,  240 = 2 × 2 × 2 × 2 × 3 × 5


4. Draw a factor tree to show the prime factorization of 42, 96 and 108.

Prime Factorization - Factor Tree Method

Therefore, 42 = 2 × 3 × 7

              108 = 2 × 2 × 3 × 3 × 3

               96 = 2 × 2 × 2 × 2 × 2 × 3.


Worksheet on Methods of Prime Factorization:

I. Prime factorize the following numbers using division method.

(i) 18

(ii) 125

(iii) 512

(iv) 144

(v) 360

(vi) 256

(vii) 96

(viii) 80

(ix) 625

(x) 169


Answer:

I. (i) 2 × 3 × 3

(ii) × 5 × 5

(iii) 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

(iv) 2 x 2 x 2 x 2 x 3 x 3

(v) 2 × 2 × 2 × 3 × 3 × 5

(vi) 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

(vii) 2 × 2 × 2 × 2 × 2 × 3

(viii) 2 × 2 × 2 × 2 × 5

(ix) 5 × 5 × 5 × 5

(x) 13 × 13


II. Find the prime factors using the factor tree.

(i) 66

(ii) 75

(iii) 24

(iv) 156

(v) 128


Answer:

II. (i) 2 × 3 × 11

(ii) 3 × 5 × 5

(iii) 2 × 2 × 2 × 3

(iv) 2 × 2 × 3 × 13

(v) 2 × 2 × 2 × 2 × 2 × 2 × 2


III. Copy and complete these factor trees.

Complete these Factor Trees

Answer:

III. (i) 2 × 2 × 5 × 5

(ii) 2 × 2 × 2 × 2 × 3

(iii) 5 × 19

(iv) 3 × 3 × 7


IV. Write the prime factorization of 24.

Answer: Prime factorization of 24 = 2 × 2 × 2 × 3


V. Draw a factor tree to show the prime factorisation of:

(i) 40

(ii) 98

(iii) 120


Answer:

V. (i)

Factor Tree of 40

(ii)

Factor Tree of 98

(iii)

Factor Tree of 120


VI. Write the prime factorization of 96.

Answer: Prime factorization of 96 = 2 × 2 × 2 × 2 × 2 × 3

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