# CLASS-8HIGHEST COMMON FACTOR (HCF)

Highest Common Factors (HCF)

The greatest number which is a common factor of two or more numbers is their Highest common factors (HCF) or Greatest Common Divisor (GCD)

Example –

The factors of  36 = 3 , 2, 4, 9, 12, 6, 18

The factors of 120 = 2, 3, 4, 5, 6, 8, 10, 12, 15, 24, 60, 40, 30, 20

The common factors of 36 & 120 is 2, 3, 4, 6, 12 and of these 12 is the Greatest Common Divisor (GCD) or Highest Common Factor (HCF)

So, the HCF of 36 & 120 is 12

For your information, there are two types of methods of finding HCF –

1) By Prime Factorization method,

2) By Division method

There are step by step methods of finding HCF by Prime Factorization methods are given below for better understanding –

Step.1)  Express each Number as a product of prime numbers

Step.2)  Find the common factors of the numbers

Step.3)  Find the product of the common factors, this product is the required HCF

Example.)  Find the HCF of  120, 180, 160

Answer)   120 = 2 X 60 = 2 X 3 X 20

= 2 X 3 X 2 X 10

= 2 X 3 X 2 X 2 X 5

160 = 2 X 80 = 2 X 2 X 40

= 2 X 2 X 2 X 20

=  2 X 2 X 2 X 2 X 10

= 2 X 2 X 2 X 2 X 2 X 5

180 = 2 X 90 = 2 X 2 X 45

= 2 X 2 X 3 X 15

= 2 X 2 X 3 X 3 X 5

The common factors of 120, 160 & 180 is  2, 2, 5

So the desired HCF is = 2 X 2 X 5 = 20

There are step by step methods of finding HCF by Division methods are given below for better understanding –

Step.1) We have to divide the bigger number ( consider as Dividend ) by the smaller number ( consider as Divisor ).

Step.2) If there is no reminder then the smaller number is the HCF. If there is a reminder, take it as the new divisor and take the previous divisor as the new dividend.

Step.3) we have to continue Step.2 till there is no remainder, the last divisor is the required HCF.

Now, to find the HCF of three or more numbers, take the following steps –

Step.1) Find the HCF of any two of the numbers

Step.2) Find the HCF of the third number and the HCF obtained in Step.1

This is the HCF of the three numbers, we have to continue similarly for more given numbers.

Example.) find the HCF of the following 48, 120 & 216 by division  method

Answer)  As per the given condition, we have to find the HCF of the tree numbers 48, 120, & 216

In three numbers 48, 120, & 216 here the smallest number is 48 and the biggest number is 216, so we will consider 48 as divisor and 216 as dividend

48 ) 216 ( 4

192

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24 ) 48 ( 2

48

----------

0

Here HCF of 48 & 216 is 24. Now we will consider obtained result 24 as the divisor and will consider 120 as a dividend which is the rest number from the given three numbers.

24 )  120 ( 5

120

-----------

0

As we can observe that, here HCF of the 24 & 120 is 24

Hence, the HCF of the three given number is 24     (Ans.)