# Highest Common Factor

Highest common factor (H.C.F) of two or more numbers is the greatest number which divides each of them exactly.

Highest Common Factor (HCF) is also called Greatest Common Divisor (GCD) or Greatest Common Factor (GCF).

Now we will learn about the method of finding highest common factor (H.C.F).

Steps 1:

Find all the factors of each given number.

Step 2:

Find common factors of the given number.

Step 3:

The greatest of all the factors obtained in Step 2, is the required highest common factor (H.C.F).

For Example:

1. Find the highest common factor (H.C.F) of 6 and 9.

Factors of 6 = 1, 2, 3 and 6.

Factors of 9 = 1, 3 and 9.

Therefore, common factor of 6 and 9 = 1 and 3.

Highest common factor (H.C.F) of 6 and 9 = 3.

Therefore, 3 is H.C.F. or G.C.D. greatest common divisor of 6 and 9.

H.C.F. or G.C.D. of given numbers is the greatest number which divides all the numbers without leaving a remainder.

2. Find the highest common factor (H.C.F) of 6 and 8.

Factors of 6 = 1, 2, 3 and 6.

Factors of 8 = 1, 2, 4 and 8.

Therefore, common factor of 6 and 8 = 1 and 2.

Highest common factor (H.C.F) of 6 and 8 = 2.

Therefore, 2 is H.C.F. or G.C.D. greatest common divisor of 6 and 8.

3. Find the highest common factor (H.C.F) of 14 and 18.

Factors of 14 = 1, 2, 7 and 14.

Factors of 18 = 1, 2, 3, 6, 9 and 18.

Therefore, common factor of 14 and 18 = 1 and 2.

Highest common factor (H.C.F) of 14 and 18 = 2.

Note: The highest common factor or HCF of two or more numbers is the greatest number that divides exactly the given numbers.

4. Find the highest common factor (H.C.F) of 15 and 10.

Factors of 15 = 1, 3, 5 and 15.

Factors of 10 = 1, 2, 5 and 10.

Therefore, common factor of 15 and 10 = 1 and 5.

Highest common factor (H.C.F) of 15 and 10 = 5.

5. Find the highest common factor (H.C.F) of 12 and 18.

Factors of 12 = 1, 2, 3, 4, 6 and 12.

Factors of 18 = 1, 2, 3, 6, 9 and 18.

Therefore, common factor of 12 and 18 = 1, 2, 3 and 6.

Highest common factor (H.C.F) of 12 and 18 = 6 [since 6 is the highest common factor].

6. Find the highest common factor (H.C.F) of 48 and 32.

Solution:

Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48

Factors of 32 = 1, 2, 4, 8, 16 and 32

Therefore, the common factors are 1, 2, 4, 8 and 16.

The highest common factor is 16.

Thus, highest common factor (HCF) of 48 and 32 is 16.

The common factors can be represented using venn diagram as given below.

7. Find the highest common factor (H.C.F) of 24 and 36.

Factors of 24 = 1, 2, 3, 4, 6, 8, 12 and 24.

Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Therefore, common factor of 24 and 36 = 1, 2, 3, 4, 6 and 12.

Highest common factor (H.C.F) of 24 and 36 = 12.

8. Let us find the HCF of two numbers 15 and 18.

The factors of 15 are 1, 3, 5, 15

The factors of 18 are 1, 2, 3, 6 ,9, 18

The common factors of 15 and 18 are 1, 3.

Clearly, the highest of these common factors is 3.

Therefore, 3 is the HCF of 15 and 18.

We can find the Highest Common Factor (HCF) by the following two methods:

I: Prime Factorization Method

II: Division Method

## I: Prime Factorization Method:

Working Rules to find the HCF by Prime Factorization Method:

Step I: Find the prime factors of each of the given number by the prime factorization method.

Step II: Multiply all the 'common' prime factors to get the HCF of the given numbers.

1. Find the HCF of 72 and 48.

Solution:

Find the prime factor of both the numbers

Prime factors of

72 = 2 × 2 × 2 × 3 × 3

48 = 2 × 2 × 2 × 2 × 3

Hence, the common factors are 2, 2, 2 and 3.

The required HCF = 2 × 2 × 2 × 3 = 24

Note: If one out of the two given numbers is a factor of the other, then the smaller number is the required HCF of the given numbers.

For Example:

HCF of 6 and 36 is 6.

Since 6 is the factor of 36.

## II: Division Method:

Working Rules to find the HCF of Two Numbers by Division Method:

Step I: Divide the greater number by the smaller one.

Step II: Divide the divisor by the remainder.

Step III: Continue the steps I to II till the remainder becomes zero.

1. Find the HCF of 198 and 360 using the long division method.

Solution:

Here, the last divisor is 18.

So, the HCF of 198 and 360 = 18.

2. Find the HCF of 144 and 180 by using the long division method.

Hence, the HCF of 144 and 180 is 36.

Working Rules to find the HCF of Three or More Numbers by Division Method:

Step I: : Find the HCF of any two numbers.

Step II: Find the HCF of third number and the HCF obtained in step I.

1. Find the HCF of 6, 8 and 12.

Solution:

Step I: Finding the HCF of 6 and 8

Step II: Finding the HCF of 2 and 12

Hence, the HCF of 6, 8 and 12 is 2.

Note: We can verify the result by the factorisation method also.

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