Prime Factorization and Its Uses

The method of representation of a number as the product of all its prime factors is known as prime factorization. With the advent of chapters like the Greatest Common Factor (GCF) and the Least Common Multiple (LCM), the topic of prime factorization is introduced to us. Prime factorization is of utmost significance for the calculation of the LCM and the GCF. It is also used in the field of cryptography. The world today is witnessing a revolution in the form of the coming up of cryptocurrencies like Bitcoin, Solana, Etherium, etc. These platforms are based on cryptography which basically means the protection of useful data with the help of codes. The method of prime factorization facilities in the process of cryptography. In this article, we will discuss prime factorization and how it helps us to find out the Greatest Common Factor (GCF) with the help of examples.

What Do You Mean by the Greatest Common Factor?

As the name suggests, the Greatest Common Factor between two or more numbers is the greatest factor that divides both the numbers wholly. It is also commonly referred to as the Highest Common Factor (HCF), the Highest Common Divisor (HCD), or the Greatest Common Divisor (GCD). This topic is very easy to understand and is very helpful in the advanced levels of math. You will need the Greatest Common Factor almost all the time during the simplification of two or more fractions. There are numerous methods to find out the GCF between two or more numbers. However, in this module, we will learn how to find out the GCF using the method of prime factorization.

Obtaining Highest Common Factor With the Method of Prime Factorisation 

It is one of the easiest methods to find out the GCF. Prime factorization is also employed to find out the LCM of two or more given numbers. Follow the steps given below to find out the GCF with the help of prime factorization method:

• Suppose, it is given to find out the GCF between two numbers.
• Write the prime factors of the numbers given. Example: Prime factors of 10 are 2 and 5 which is also written as 2 *5.
• Pick out the common factors from both the given numbers.
• The product of the common factors is the Greatest Common Factor between both the numbers.

Solved Examples of GCF using the Method of Prime Factorisation

1. Find out the Greatest Common Divisor between 400 and 800.

Solution: 400 = 2* 2 * 2 * 2 * 5 * 5

               800 = 2*  2 * 2 * 2 * 2 * 5 * 5

               Common factors between 400 and 800 are 2, 2, 2, 2, 5 and 5.

               Thus, the GCF between 400 and 800 is 400.

1. Find out the Greatest Common Factor between 52, 62 and 122.

Solution: 52 = 2 * 2 * 13

               62 = 2 * 31

               122 = 2 * 61

               Common factor between 52, 62 and 122 is 2

               Thus, the GCF between 52, 62 and 122 is 2.

1. Find out the Greatest Common Factor between 39, 69 and 129.

Solution: 39 = 3 * 13

               69 = 3 * 23

               129 = 3 * 43 

               Common factor between 39, 69, and 129 is 3.

               Thus, the GCF between 39, 69, and 129 is 3.

An important thing that needs to be observed in the method of prime factorization is that the factors of the numbers should all be prime numbers. Example: 1, 2, 4, and 16 are the factors of 16. However, during the process of prime factorization, 2 will be the only prime number that will come into consideration. 

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