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Prime Factorization

A number in general has two diverging aspects. A number can be viewed as a factor of a larger number and at the same it is formed as product of some smaller numbers.The concepts of factor decomposition and product composition are applied at any level of Mathematical problem solving.

Prime Factorization of a number is finding the prime factors of the number. Every natural number greater than 1, is either a prime or a product of primes in a unique way.

Prime Factorization Definition

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Prime Factorization means breaking up a composite number into prime factors. The concept of factors and the idea of factorization originate from the divisibility of numbers. If a number divides another number with a remainder "0", then the second number is said to be divisible by the first number.

Solved Example

Question: Find the prime factors of 12.

Solution:
When 12 is divided by 3, the quotient is 4 and the remainder is 0. So we can say that 12 is divisible by 3.

Hence prime factors of 12,

12 = 2 x 2 x 3 = 22 x 3.
 

What is Prime Factorization?

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Prime factorization is the process of writing a composite number as a product of its prime factors. A composite number written as a product of prime numbers is known as the prime factorization of the number. In prime factorization the order of numbers does not matter. This makes there is only one prime factorization for a number.

Solved Example

Question: Prime factorization of 18.

Solution:
Prime factors of 18,

18 = 2 x 3 x 3  = 2 x 32

or 3 x 3 x 2 = 32 x 2

or 3 x 2 x 3 = 32 x 2.
 

How to do a Prime Factorization of a Number?

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The divisibility test can be used to identify the prime factors of a number. Divisibility test help to find the factors of a number. The following table details the divisibility tests for some of the prime numbers.

Divisible by

                               Divisibility Test
      2  The ones digit is 0, 2, 4, 6 or 8. In other words the number should be an even number.
      3  The sum of the digits is divisible by 3.
Example: 231
231 = 2 + 3 + 1 = 6, 6 is divisible by 3.
So 231 is divisible by 3.
      5  The ones digit is 0 or 5
Example: 120
Here ones digit is zero
So 120 is divisible by 5.
      7 Double the ones digit and subtract that from the sum of the rest of the digits.Repeat the procedure till a two digit number is arrived. If the two digit number got is divisible by 7, then the given number is divisible by 7.
    11 Subtract the sum of the digits in odd places from the sum of the digits in the even places. If the difference obtained is 0 or divisible by 11, then the given number is divisible by 11.


Solved Example

Question: Write -

Prime Factorization of 72


Solution:
By divisibility test:

Since 72 is an even number, is divisible by 2

72 = 2 x 36.

Continuing the division with 2,

36 = 2 x 18 and 18 = 2 x 9.

9 = 3 x 3

Hence prime factorization of 72,

72 = 2 x 2 x 2 x 3 x 3 = 23x 32
 

Prime Factorization Tree

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Another method of doing prime factorization is using a prime factorization tree also commonly called a factor tree. The factor tree is formed by branching out the factors at each level of prime division.

Solved Example

Question: Use factor tree to find -

Prime Factorization of 245


Solution:
The ones digit is 5, so the number is divisible by 5.

Dividing 245 by 5 we get the quotient as 49.

again 49 is divisible by 7 and 49 = 7 x 7.  

The factor tree can be extended and completed as follows:

Prime Factorization Tree
                                                                        
The process is stopped at that level when the quotient on division by a prime is also a prime.

Hence the prime factorization is 245,

245 = 5 x 7 x 7 = 5 x 72.
 

Prime Factorization in Finding the GCF

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Prime factorization is used in finding the greatest common factor of two or more numbers. The greatest common factor of two or more numbers is the product of all common factors of the numbers.                   

Solved Example

Question: Find the greatest common factor of the numbers 24 and 60.

Solution:
The Prime factorization of the two numbers 24 and 60 is shown here::
 
GCF Using Prime Factors

Prime factors of 24 = 2 x 2 x 2 x 3

Prime factors of 60 = 2 x 2 x 3 x 5

Hence GCF of 24 and 60 = 2 x 2 x 3 = 12   
 

Prime Factorization Examples

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Below you could see some example of prime factorizations:

Solved Examples

Question 1: Write -

Prime Factorization of 36


Solution:
By divisibility test:

36 is even number, a number with 6 as ones digit.

So 2 is a prime factor of 36. On division by 2 the quotient is 18.

=> 36 = 2 x 18

Again 18 is an even number and can be written as 18 = 2 x 9.

Now we have 36 = 2 x 2 x 9,  9 is divisible by 3 and 9 = 3 x 3

Hence the prime factorization of 36,

36
= 2 x 2 x 3 x 3

If the repeated factors are replaced by powers, then

36 = 22 x 32.
 

Question 2: Write -

Prime Factorization of 48


Solution:
By divisibility test:

48 being an even number is divisible by 2.

48 = 2 x 24

Continuing the division with 2,

24 = 2 x 12 and 12 = 2 x 6 and 6 = 2 x 3.

Hence the prime factorization for 48 is

48 = 2 x 2 x 2 x 2 x 3 = 24 x 3.

 

Question 3: Write -

Prime Factorization of 75


Solution:
By divisibility test:

As the ones digits is 5, this number is divisible by 5. 

Dividing 75 by 5 the quotient is 15,  75 = 5 x 15.

15 can be factored in a similar manner as 15 = 3 x 5.

Hence the prime factorization of 75 is

75 = 5 x 5 x 3 = 52x 3