Prime factorization

Prime factorization is the process of breaking down a composite number into a product of its prime factors. It involves dividing the number by prime numbers until only prime numbers remain.

• 1. Definition of a Prime Number
• 2. What is Prime Factorization
• 3. What are Prime Factors
• 4. Prime Factorization Formula
• 5. List of Prime numbers from 1 to 100?
• 6. Difference between Factors and Prime Factors?
• 7. How to find the Prime Factorization of a Number
• 8. Different methods to find Prime Factorization
• 9. Difference between Prime numbers and Composite numbers
• 10. Difference between Co-Prime and Twin Prime numbers
• 11. A video Explaining this Concept
• 12. Solved Examples related to Prime Factorization
• 13. Frequently Asked Questions (FAQs)
• 14. Real-life Application of Prime Factorization
• 15. Conclusion

1. Definition of a Prime Number:

A Prime number is a natural number greater than 1 with exactly two distinct positive factors, i.e., 1 and itself. Examples include 2, 3, 5, 7, and so on.

2. What is Prime Factorization:

Prime factorization is a method of expressing a Composite number as a product of its prime factors. It's like breaking down a number into its building blocks, each block being a prime number.

For example, If we take the number 36.
Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
But we can see that all the factors of 36 are not Prime factors
So, we will convert those non-prime factors again as a multiple of prime factors.
like
4 = 2 x 2,
6 = 2 x 3,
9 = 3 x 3,
12 = 2 x 2 x 3,
18 = 2 x 3 x 3,
36 = 2 x 2 x 3 x 3

3. What are Prime Factors:

Prime factors are the prime numbers that, when multiplied together, give the original number.

Factors of a Natural number can be non-prime, too, but we have to consider prime numbers only.
For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, but prime factors are 2 and 3 only.
Let us explore the prime factorization chart of some random numbers in the table given below:

4. Prime Factorization Formula:

Any composite number can be expressed as the product of powers of prime numbers, and this way of writing the composite number is called prime factorization. The prime factorization formula for any number is given as:
$\bold{N = x_1^{a_1}.x_2^{a_2}.x_3^{a_3}.....x_k^{a_k}}$
where,

• N = Any Composite Number
• $x_1, x_2, x_3,...., x_k$ are Prime Factors of N respectively.
• $a_1, a_2, a_3,...., a_k$ are respective exponents (powers) of Prime Factors of N.

5. List of Prime numbers from 1 to 100:

There are 25 prime numbers within the range from 1 to 100.
$\fcolorbox{red}{aqua}{$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and, 97$}$

6. Difference between Factors and Prime Factors:

Factors are the numbers that divide another number exactly, whereas prime factors are the factors that are prime numbers.
For example, the factors 12 are 1, 2, 3, 4, 6, and 12, while the prime factors are 2 and 3 only.

7. How to find the Prime Factorization of a Number:

We can find the prime factorization of a Number by the most commonly used two methods:
$\bold{(i)}$ $\fcolorbox{red}{aqua}{$Division \space Method$}$
$\bold{(ii)}$ $\fcolorbox{red}{aqua}{$Factor \space Tree \space Method $}$

8. Different methods to find Prime Factorization:

(i) Prime Factorization by Division Method:

This method is used to find the factors of large value numbers.
This method divides the number by a series of prime numbers.
Using an example, let us learn how to find the prime factors using the division method.
$\bold{Example:}$ Find the prime factors 56 using the division method.
$\bold{Solution:}$

• $\bold{Step \space 1:}$ Divide the number by the smallest prime number such that it divides the number completely. Here, we can divide 56 by 2 to get 28.
  
• $\bold{Step \space 2:}$ Now the quotient is 28, and again, we divide the quotient by the smallest prime number. So here, we can divide 28 by 2 to get 14.
  
• $\bold{Step \space 3:}$ Now we must repeat step 2 until we get 1 in the quotient. Since the quotient in the above step is 14. So here, we can divide 14 by 2 to get 7.
  
• $\bold{Step \space 4:}$ Now the quotient is 7; if we divide 7 by 7, we will get 1 as the quotient, and the method will terminate here. So, the prime factors of 56 are 2 and 7.
  Finally, we can write prime factorization of 56 = 2 x 2 x 2 x 7.
  

(ii) Prime Factorization by Factor Tree Method:

In this method, factors of a number are found initially, which are further factorized until we get the prime factors. In this way, a tree with its branches of factors is formed. Now, we will learn how to use the tree method with the help of an example.
$\bold{Example:}$ Find the prime factorization of 650 using the Factor Tree method.
$\bold{Solution:}$

• $\bold{Step \space 1:}$ Consider the given number 650 on the top of the tree.
• $\bold{Step \space 2:}$ Write down the above number as a multiple pair of two composite numbers. Here, we can write 650 as a multiple of 65 and 10.
• $\bold{Step \space 3:}$ We need to factorize both composite numbers obtained in the above step, i.e., 65 = 5 x 13 and 10 = 2 x 5.
• $\bold{Step \space 4:}$ Repeat step 3 until we get all the factors as prime factors. In the above step, we got all factors as prime factors, and we will terminate this method here.
  So, Prime factorization of 650 = 2 x 5 x 5 x 13.

9. Difference between Prime numbers and Composite numbers:

$\bold{Prime \space numbers}$ have exactly two factors i.e., 1 and itself.
Examples of prime numbers are 2, 3, 5, 7, ....
$\bold{Composite \space numbers}$ have more than two factors. We can say that Prime numbers are like the atoms of maths, while composite numbers are their combinations.
Examples of Composite numbers are 12, 18, 45, 100, ....

10. Difference between Co-Prime and Twin Prime numbers:

$\bold{Co \space Prime \space Numbers:}$ when the HCF of a pair of Prime numbers is 1 or if a pair of prime numbers has no other common factor other than 1, then such prime numbers are called Co-prime or Relatively Prime numbers.
For example: (2, 3), (3, 5), (7, 13) .... are examples of Co Prime numbers.
$\bold{Twin \space Prime \space Numbers:}$ It is defined as when the difference of a pair of prime numbers is 2, then such prime numbers are called Twin Prime numbers.
For example: (3, 5), (5, 7), (11, 13) .... are examples of Twin Prime numbers.

11. A video Explaining this Concept:

12. Solved Examples related to Prime Factorization:

$\bold{Example \space 1:}$ Find the prime factorization of 72 using the division method.
$\bold{Solution:}$
After repeated division to obtain all prime factors of 72, we get: 72 = 2 x 2 x 2 x 3 x 3

$\bold{Example \space 2:}$ Find the prime factorization of 48 using the factor tree method.
$\bold{Solution:}$
After repeated division to obtain all prime factors of 48, we get the factor tree: 48 = 2 x 2 x 2 x 2 x 3

13. Frequently Asked Questions (FAQs):

Is 1 a prime number?

One is neither a prime nor a composite number.

Which is the smallest prime number?

The smallest prime number is 2.

Does any prime number even exist?

The only even number which is prime is 2.

Which is the largest prime number?

The largest prime number is still not known because it tends to infinity.

Can we find the prime factors of negative numbers?

No, because prime factors are not defined for negative numbers.

Are there any shortcuts or tricks for finding prime factorizations quickly?

While there are no universal shortcuts, this Prime Factorization Calculator can help u to find the prime factors of any number quickly.

What is prime factorization, and why is it important?

Prime factorization is the process of expressing a composite number as the product of its prime factors. It's essential because it helps simplify fractions, find the greatest common divisor (GCD) and least common multiple (LCM), and solve various mathematical problems.

What are the practical applications of prime factorization outside of mathematics?

Prime factorization is used in cryptography for encryption and decryption algorithms, in computer science for data compression and error correction, and finance for calculating interest rates and loan terms.

14. Real-life Application of Prime Factorization:

In real life, it is majorly used in the following fields: -

• Cryptography
• Computer Algorithms
• Finding least common multiple (LCM) and greatest common divisor (GCD)

15. Conclusion:

Prime factorization is a powerful tool that plays a crucial role in number theory, cryptography, and various other fields. By understanding the concepts and techniques discussed in this blog, you'll be equipped to tackle complex mathematical problems and appreciate the beauty of prime numbers in the world around us. Keep exploring, keep learning, and let the magic of prime factorization inspire you on your mathematical journey!

If you have any suggestions regarding the improvement of the content of this page, please write to me at My Official Email Address: [email protected]

$\fcolorbox{black}{lightpink}{\color{blue}{Click here to Ask any Doubt}}$
Are you Stuck on homework, assignments, projects, quizzes, labs, midterms, or exams?
To get connected to our tutors in real-time. Sign up and get registered with us.

$\color{red} \bold{Related \space Pages:}$
Prime Factorization Calculator
Factors of a Number Calculator
Divisibility Rules Blog