The greatest common factor is also called as the Greatest Common Divisor GCD or Highest Common Factor HCF. It is the factor that divides two numbers. The greatest common factor can be found by listing of the prime factors of each of the numbers; then these factors need to be multiplied by the numbers that both have in common. Incase there are no common prime factors then it means that the greatest common factor is 1.

For Example, the greatest common factor of 9 and 12 can be denoted as
The Prime Factors of 9 which are 3 x 3
The prime factors of 12 which are 2 x 2 x 3
Therefore the common factor is 3 which make the GCF 3
The Greatest common factor is used for finding the largest factor which is used to divide two numbers.
In order to find the GCF of 15 and 60
The prime factors of 15 are 5 x 3
The prime factors of 60 are 5 x 3 x 2 x 2
Therefore as 5 and 3 are common, the GCF is 3 x 5 or 15

When Finding the GCF there are 2 Methods: a) First, the common factors need to be compared. Then the sets of factors need to be compared till the biggest number which is in both sets is identified. b) The second method is the prime number method. In this, the factors of each number first need to be broken into the prime numbers. Then common prime factors need to be picked out. After which all the common prime factors need to be multiplied, and that is how you arrive at the greatest common factor.
The Greatest Common Factor in a polynomial needs to be checked as factoring of the polynomial is easier when the GCF is factored. The terms get less cumbersome. Incase the GCF has a variable, it gets easier. Incase there is a polynomial equation 6y4-12y3+4y2 these are the steps to be followed:
The terms are to be broken into the prime factors which are
(3 x 2 x y x y x y x y)- (2 x 2 x 3 x y x y x y) + (2 x 2 x y x y)
Then the factors which are in every single term should be determined. In the above example, the common factors are 2y2
The GCF is then to be factored out and placed in front of the parenthesis and then the remnants are to be grouped inside, so it will be 2 x y x y (3 x y x y – 2 x 3 x y + 2)
Then the 2y2 is to be multiplied to each of the factors inside the parenthesis to check and see if the GCF calculated is correct.
This is how GCF is calculated. The GCF is used for all kinds of calculations including for working with fractions. Therefore, this is a concept which needs to be thoroughly understood.

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