Prime Factorization

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Introduction to Prime Factorization

Prime factorization is a process of factoring a number into its prime numbers i.e the factors will be prime numbers. In other words, it is breaking down numbers into its prime factors. Here, we will discuss the concepts of prime factors, what is prime factorization and different prime factorization methods with solved examples. After going through the article students should be able to understand prime factors of a number and how to find the prime factors of a number easily.


The easy and simplest method to find the prime factorization of a number is to keep on dividing the original number by prime factors until the remainder equal to 1. For example, the prime factor of number 20 we get, 20/2 = 10, 10/2 = 5, 5/5 = 1. Since we received the remainder as 1 and it cannot be further factorised. Therefore, 20 = 2 x 2 x 5, where 2 and 5 are prime factors.


What is Prime Factorization?

Prime factorization is a method of finding the prime factors of a number, such that the original number is divisible by these factors. As we know that a composite number has more than two factors. Therefore, this method is only applicable to composite numbers and not for prime numbers.

For example, the prime factors of 120 will be 2, 2,2,3 and 5 as 2 × 2 × 2 × 3 5 = 126 and 2, 3, 5 are prime numbers.

Few more example of Prime factorization :

6 is 2 x 3 

18 is 2 x 3 x 3 = 2 x 3² 

25 is 5 x 5 = 5²

20 is 2 x 2 x 5 = 2² x 5

The prime numbers when multiplied by any other natural numbers or whole numbers (except 0), will give composite numbers. So basically prime factorization is performed on the composite numbers to factorize them and find their prime factors. This method is also used to find the HCF( Highest Common Factors) and LCM ( Least Common Factors) 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 together it will give the actual number. Also, we can say that the prime factors divide the number completely. It is similar to factors of a number and considering only the prime numbers among the factors. For example, the prime factors of 10 will be 2 and 5, the prime factors of 15 will be 3 and 5 etc.


Prime Factorization Methods

There are two methods to find the prime factorization of a number:

(i) Division Method

(ii) Factor Tree Method


(i) Division Method

In the division method, we keep on dividing numbers until the quotient becomes 1. 

Following are steps to find the prime factors of a number using the division method:

Step 1: First divide the given number by the smallest prime number. In this case, the smallest prime number should be taken in such a way that it divides the number exactly.

Step 2: Again, divide the quotient by the possible smallest prime number.

Step 3: Repeat the same process, until the quotient becomes 1.

Step 4: Finally, multiply all the prime factors to get the original number.

Let’s understand this with an example


Question: Using the division method, find the prime factorization of 370.

Step 1: Divide 370 by the least prime number i.e. 2.

          So, 370 ÷ 2 = 185

Step 2: Again Divide 185 with the least prime number (which is 5).

          Now, 185 ÷ 5 = 37

Step 3: As we know 37 is a prime number, divide it by itself to get 1.

      So, 37 ÷ 37 = 1

Now we multiply all the prime factors to get the original number i.e 370. Hence, the prime factors of 370 will be 2 x 5 x 37


Factor Tree 


Factor Tree Method

Following are the steps to find the prime factorization of the given number using the factor tree method. 

Step 1: Consider the given number as the base of the tree

Step 2: Write down the pair of factors in the branches form of a tree

Step 3: Again factorize the composite factors, and write down the factors pairs in the branches form of the tree.

Step 4: Repeat the step, until you have found the prime factors of all the composite factors

In the factor tree, the factors of a number are found and then those numbers are further factorized until we reach the closure. Consider two numbers to find the factors of 60 and 282 using a factor tree. See the given below diagram to understand the concept.

[Image will be Uploaded Soon]

In the above figure, we can see that the number 60 is first factored into two numbers i.e. 6 and 10. Again, 6 and 10 is factored into two numbers to get the prime factors of 6 and 10.

6 = 2 x 3

and 10 = 2 x 5

If we write all the prime factors of 60 altogether, then we will get 

60 = 10 x 6 = 2 x 3 x 2 x 5

The same is the case for number 282, such as;

282 = 2 x 141 = 2 x 3 x 47

Therefore in both cases, a tree structure is formed.


Solved Examples:

1.Find the prime factors of 500.


Steps

Prime Factors

Division

Step 1 :Divide by 2 

2

500 ÷ 2 = 250

Step 2 :Divide by 2 

2

250 ÷ 2 = 125

Step 3 : Divide by 5

5

125 ÷ 5 = 25

Step 3 : Divide by 5

5

25 ÷ 5 = 5

Step 3 : Divide by 5

5

5 ÷ 5 = 1


Hence prime factorisation of 500 is 2² x 5³ 


2.Find the prime factorisation of 1050


Steps

Prime Factors

Division

Step 1 : Divide by 2 

2

1050 ÷ 2 = 525

Step 2 : Divide by 5

5

525 ÷ 5 = 105

Step 3 : Divide by 5

5

105 ÷ 5 = 21

Step 4 : Divide by 21

21

21 ÷ 21 = 1


Hence prime factorisation of 1050 is 2 x 5² x 21


Facts about Prime Factorisation:

  • Every whole number is greater than 1 and it can be represented as either prime or can be written as a product of prime numbers.

  • This prime factorisation is a unique  number, apart from the order in which we write the prime factors.

FAQ (Frequently Asked Questions)

1. Define Prime Factorization.

Ans: Prime factorization is the process of finding the prime numbers. When we multiply together these prime factors we will get the original number. For example, the prime factors of 8 are 2 x 2 x 2 . This can also be written as 2³.

2. Write the Prime Factorisation of 36?

Ans: The number 36 can be written as the product of 9 and 4. Further, it can be written in prime factorisation form as 3² x 2²

3. Define Factor Tree.

Ans: A factor tree is a diagram like a tree structure used to determine the prime factors of a natural number It is one of the methods to find the prime factorisation of a number.