HCF - Highest Common Factor

The greatest number which divides each of the two or more numbers is called HCF or Highest Common Factor. It is also called the Greatest Common Measure(GCM) and Greatest Common Divisor(GCD). HCM and LCM are two different methods, whereas LCM or Least Common Multiple is used to find the smallest common multiple of any two or more numbers.

Example: The Highest common factor of 60 and 75 is 15 because 15 is the largest number which can divide both 60 and 75 exactly.

We can find the HCF of any given numbers by using two methods:

  • by prime factorization method
  • by division method

Let us discuss these two methods one by one in this article.

Also, read:

HCF By Prime Factorization Method

Follow the below-given steps to find the hcf of numbers using prime factorization method.

Step 1: Write each number as a product of its prime factors. This method is called here prime factorization.

Step 2: Now list the common factors of both the numbers

Step 3: The product of all common prime factors is the HCF( use the lower power of each common factor)

Let us understand with the help of examples.

Example 1: Evaluate the HCF of 60 and 75.

Solution:

Write each number as a product of its prime factors.

22 x 3 x 5 = 60

3 x 52 = 75

The product of all common prime factors is the HCF( use the lowes power of each common factor)

The common prime factors in this example are 3 & 5.

The lowest power of 3 is 3 and 5 is 5.

So, HCF = 3 x 5 = 15

Example 2: Find the HCF of 36, 24 and 12.

Solution:

Write each number as a product of its prime factors.

22 x 32 = 36

23 x 3 = 24

22 x 3 = 12

The product of all common prime factors is the HCF( use the lowes power of each common factor)

The common prime factors in this example are 2 & 3.

The lowest power of 2 is 22 and 3 is 3.

So, HCF = 22 x 3 = 12

Example 3: Find the HCF of 36, 27 and 80.

Solution:

Write each number as a product of its prime factors.

22 x 32 = 36

33 = 27

24 x 5 = 80

The product of all common prime factors is the HCF( use the lowes power of each common factor)

The common prime factors in this example are none.

So, HCF is 1.

HCF By Division Method

You have understood by now the method of finding the highest common factor using prime factorization. Now let us learn here to find HCF using division method. Basically division method is nothing but dividing the given numbers simultaneously to get the common factors between them. Follow the steps mentioned below to solve problems of hcf.

  • Step 1: Write the given numbers horizontally, in a sequence, by separating it with commas.
  • Step 2: Find the smallest prime number which can divide the given number. It should exactly divide the given numbers. (Write in the left side).
  • Step 3: Now write the quotients.
  • Step 4: Repeat the process, until you reach the stage, where there is no coprime number left.
  • Step 5: We will get the common prime factors as the factors in the left-hand side divides all the numbers exactly. The product of these common prime factors is the HCF of the given numbers.

Let us understand the above mentioned steps to find the hcf by division method with the help of examples.

Problem 1: Evaluate the HCF of 60 and 75

HCF Example1

As we can note that the mentioned prime factors in the left side divide all the numbers exactly and so, they all are common prime factors. We have no common prime factor for the numbers remained at the bottom.

So, HCF = 3 × 5 = 15.

Example 2: Find out HCF of 36, 24 and 12

highest common factors Example

HCF = 2 × 2 × 3 = 12.

Example 3: Find out HCF of 36, 24 and 48.

HCF by division method

HCF = 2 × 2 × 3 = 12.

HCF by Shortcut method

Steps to find the HCF of any given numbers.

  • Step 1: Divide larger number by smaller number first, such as;

Larger Number/Smaller Number

  • Step 2: Divide the divisor of step 1 by the remainder left.

Divisor of step 1/Remainder

  • Step 3: Again divide the divisor of step 2 by the remainder.

Divisor of step 2/Remainder

  • Step 4: Repeat the process until the remainder is zero.
  • Step 5: The divisor of the last step is the HCF.

How to find the HCF of 3 numbers

1) Calculate the HCF of 2 numbers.
2) Then Find the HCF of 3rd number and the HCF found in step 1.
3) The HCF you got in step 2 will be the HCF of the 3 numbers.

The above steps can also be used to find the HCF of more than 3 numbers.

HCF Examples

Here are a few more example to find the highest common factors.

Example 1: Find out HCF of 60 and 75.

Finding HCF by Shortcut method

So, the HCF of 30 and 45 is 15.

Example 2: Find out HCF of 12 and 36.

HCF by shortcut method

So, HCF of 12 and 36 = 12

Example 3: Find out HCF of 9, 27, and 48

Take any two numbers and find out their HCF first. Say, let’s find out HCF of 9 and 27 initially.

highest common factors by shortcut method

So, HCF of 9 and 27 = 9

HCF of 9 ,27, 30

= HCF of [(HCF of 9, 27) and 30

= HCF of [9 and 30]

Shortcut method for hcf

Hence, HCF of 9 ,27, 30 = 3

Example 4: Find out HCF of 5 and 7

HCF Calculation

Hence HCF of 5 and 7 = 1

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