HCF

HCF - How to Calculate HCF

What does HCF mean?

  • The largest common factor of all the given numbers is known as the Highest Common Factor of the numbers.

  • The highest number that can be divided exactly into two or more numbers without any remainders.

  • It is also known as the Greatest Common Divisor (GCD).

  • The most easy way to find the HCF of two or more given numbers is to create a factor tree.


Here are a Few Highest Common Factor Examples : Images to be uploaded soon


The above picture shows how you can calculate the H.C.F. of 90 and 30 using the prime factorization method.


How to Calculate HSF(highest common factor) of Any Two or More Given Numbers?

There are three methods of how to find the highest common factor of any two or more given numbers:

  1.  Factorization Method

  2. Prime Factorization Method

  3. Division Method

     Let’s know how these methods work !


  1. FACTORIZATION METHOD 

In the hcf by factorization method,we find the greatest common factor by listing down the factors of the numbers.

Step 1: List down the factors of all the given numbers.

Step 2: Look for the highest factor among the listed factors which are common for both the given numbers.


Let us take an example:


Problem : Find the H.C.F. of 36 and 45.

Solution: 36 = 1,2,3,4,6.9,12,18,36

45 =1,3,5,9,45


Since, the highest common factor of 36 and 45 is 9, therefore H.C.F of 36 and 45 is 9.

  1. PRIME FACTORIZATION METHOD (Also known as the factor tree method)


   Let us consider an example,


Step 1: In calculating the hcf by prime factorization, we factorize the numbers into prime numbers, which is 

            known as the prime factors.

   Step 2: Start by dividing the given numbers by 2(the first prime number),go 

on dividing till you can’t divide the number any further.

 Step 3: Finally, then write the numbers as a product of the prime numbers. The           product of these common factors is the highest common factor of the given numbers.

Here’s an example for better understanding.


Find the hcf by prime factorization of the numbers 24 and 36.

Solution: To find the hcf by prime factorization ,

 



            24 = 2×2×2×3×1

36 = 2×2×3×3×1

           

Since, the common factors of 24 and 36 are 2×3×2×1= 12


           Therefore, hcf of 24 and 36 is 12.

  1. DIVISION METHOD:

In division method, we follow the following steps:

Step 1:  Treat the smallest number as the divisor,and the bigger number as the dividend.

Step 2:  Further divide the divisor by the remainder.

Step 3: Stop dividing when the remainder equals to  zero.

Here’s an example,

Find the H.C.F. of 12,18.


          Solution:

                                     12) 18 (1

                               - 12  

                                      6 )12 (2

                                      -12

                                                   0

Here, 6 is the H.C.F.


How to Find the HCF of 3 Numbers or How to Find the Highest Common Factor:

We can find the hcf of 3 numbers either by Prime Factorisation Method or by Division Method. However, the steps for finding the highest common factor remains the same as  above.

Here are a few highest common factor examples of how to find the highest common factor ,

1.Find the hcf of 3 numbers 15,30 and 90 using the Prime Factorization method.

Solution:


3

15

5

5


1


2

30

3

15

5

5


1


2

90

3

45

3

15

5

5


1



15= 5×3×1

30= 5×3×2×1

90= 3×3×2×5×1

The common factors here are 1,3,5.

Therefore,the highest common factor of the numbers,15,30 and 90 is 5×3×1=15.


2. Find the highest common factor of 184,230 and 276 by using the Division Method.

Solution:

                          

184) 230 (1

      - 184

         46    )184 (4

                  -184

                       0


The Highest Common Factor of 184 and 230 is 46.

       46) 276 (6

      -276

         0


The Highest Common Factor of 276 and 46 is 46.

Therefore , the highest common factor(H.C.F.) of 184,230 and 276 is 46.


QUESTIONS TO SOLVE:

Question 1) What is the H.C.F. of  25,30,35 and 40?

Solution: Solving by Factorisation Method,

25 = 5×5

30 = 5×3×2

35 = 5×7


Therefore, the H.C.F of the numbers 25,30,35 and 40 is 5.


Question 2) Find the highest common factor of 64 and 80 by using the prime factorization method?

Solution: Solving by Prime Factorization Method,



The factors of 64 and 80 are,


64=2×2×2×2×2×2×1

80=2×2×2×2×5×1


Therefore, the Highest Common Factor of 64 and 80 is, 2×2×2×2×1=16


Question 3) Find the highest common factor or the Greatest Common Divisor of 16 and 27.

Solution:


Therefore, the Highest Common Factor of 16 and 27 is 1.


Question 4) Find the Highest Common Factor (H.C.F.) of 12,15 and 45.

Solution: Solving the above given question using division Method,


                12) 15(1

            - 12

                     3)12(4

               -12

                          0


We come to know that 3 is the Highest Common Factor of 12 and 15


Now, let’s find the Highest Common Factor of 3 and 45.

            

              3) 45(15

                 -45

                    0

                

Therefore, the Highest Common Factor of the numbers 12,15 and 45 is 3.


Question 5) Find the H.C.F. of 136,170 and 255 by using division method.

Solution: Let us find the Highest Common Factor of 136 and 170 first.


136) 170(1

      - 136

         34)136(4

              -136

                  0

Highest Common Factor of 170 and 136 is 34.


Now, we need to find the H.C.F. of 34 and 255,


            34) 255(7

                 - 238

                      17)34(2

                  -34

                            0

The H.C.F. of 34 and 255 is 17.

Therefore, the  required highest common factor of 136,170 and 255 is 17.


Question 6) Find the H.C.F. of 160, 220, 340.

Solution:

Let us solve the question by using,

160 =2×2×2×2×2×5×1

220 =2×2×5×11×1

340 =2×2×5×17×1

So, the numbers common in the sequence are 2×2×5×1

Therefore, the H.C.F of the numbers 160,220 and 340 is 20.


IMPORTANT TERMS YOU NEED TO KNOW:

  1. FACTORS - A factor is a number which exactly divides other number without    leaving any remainder. 

        Example: 3 and 5 are factors of 15


  1. PRIME FACTORS- A prime factor is a number that does not have any factors except 1 and the number itself. 

         Example: 2,3,5,7,11,13,17 and so on.

  1. FACTOR TREE-  A diagram used to list down the prime factors of a number in a tree diagram.

  2. COMMON FACTOR - A factor is a common factor when it is a factor of two or more numbers.

        Example: Let’s find out the common factor of 6 and 3.

           First, list down the factors of both the given numbers.

        6=2×3×1

        3=3×1

        3 is a factor of 6 as well as it is a factor of 3.

        Hence, 3 is a common factor of 6 and 3.

FAQ (Frequently Asked Questions)

1. How do I Find the Highest Common Factor?

We get the highest common factor by multiplying all the factors that appear in        

both the lists. 

Example : The highest common factor of 6 and 12 is 2×3×1= 6

2. Why do we use Highest Common Factor?

We use highest common factor to simplify fractions. Highest common factor or the greatest common divisor is the greatest number that divides exactly into two numbers.