A number dividing the other number wholly is called the factor. In a layman’s language, a factor is nothing but a divisor of the number in consideration. It is a very important concept in mathematics and is used in a variety of things. We use the process of factorization in real-life situations like comparing different prices, division of any number of rows or columns in equal parts, etc. The word factor has been derived from a Latin word, the meaning of which is ‘a doer’ or ‘a performer’. In this article, we will learn in detail about the concepts of factors and the Greatest Common Factor. We will also discuss how these two concepts are related to each other.

## Some Important Properties Related to Factors

- The factors of any given number are finite in nature.
- The factors of a prime number will always be two.
- The factors of any given number will always be equal to or less than the given number.
- The arithmetic operations used to find factors are division and multiplication.

## What Do You Mean by the Greatest Common Factor?

The greatest number which is a factor of two or more numbers in consideration is called the Greatest Common Factor. This is also sometimes referred to as the Highest Common Factor or the Greatest Common Divisor. We use factors to find out the Greatest Common Factor. The list of prime factors of each number in consideration is found out to obtain the Greatest Common Factor. Now, a question will pop into your head. What is a prime factor? Let us now find out what a prime factor is.

## What Do You Mean by the Prime Factors?

Prime factors are the factors of a number that are primes. The concept of prime factors is used to find out the Greatest Common Factor as well as the Least Common Multiple. Let us discuss some of the prime factors to understand it more clearly. Examples: The prime factors of 18 are 2, 3, and 3. The prime factors of 22 are 2 and 11.

## Procedure to Find out the Greatest Common Factor Using the Prime Factors

Follow the steps mentioned below to find out the Greatest Common Factor with the help of the prime factors:

- You will be given two or more numbers. Obtain the prime factors of each number.
- Now, take out the common factors between.
- Multiply the common factors to obtain the Greatest Common Factor between the given numbers.
- If there are no common factors between the given set of numbers, the Greatest Common Factor will be 1.

## Some Solved Examples of the GCF Using the Method of Prime Factorization

1. Find out the Greatest Common Factor of the given numbers: 17, 34, and 68.

Solution: Prime Factors of 17: 17

Prime factors of 34: 2, 17

Prime factors of 68: 2, 2, 17

Common factor between 17, 34, and 68 is 17.

Thus, the Greatest Common Factor is 17.

2. Find out the Greatest Common Factor of the numbers 21, 63, and 84 using the method of prime factorization.

Solution: Prime factors of 21: 3, 7

Prime factors of 63: 3, 3, 7

Prime factors of 84: 2, 2, 3, 7.

Common factors between 21, 63, and 84 are 3 and 7.

Thus, the Greatest Common Factor of the given numbers = 3 * 7 = 21.

3. Find out the Greatest Common Factor of the numbers 61, 17, 11, and 21 using the method of prime factorization.

Solution: Prime factors of 61: 61

Prime factors of 17: 17

Prime factors of 11: 11

Prime factors of 21: 3, 7.

We see here that no factor is common in all the given numbers except the number 1.

Thus, the Greatest Common Factor between the given numbers is 1.

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