Highest Common Factor

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Factor Definition

If a number can be expressed as the product of two numbers, then each number of the product is a factor of the given number.

For example:

Factor definition explained using Factorisation of 18

Similarly, \(18\) can be written as the products of the following numbers:

\[\begin{align} 18 &= 1 \times 18 \\ 18 &= 3 \times 6 \end{align}\]

Thus, the factors of \(18\) are \(1,2,3,6,9\) and \(18\).


Common Factor Definition

A common factor of two or more numbers is a factor that is common to both of the given numbers.

For example, the common factors of two numbers \(18\) and \(27\) can be found by listing the factors of both numbers.

Common factor definition explained using Common factors of 18 and 27

Thus, the common factors of \(18\) and \(27\) are \(1,3\) and \(9\).


HCF (Highest Common Factor) Definition

  • The HCF (Highest Common Factor) of two or more numbers is the highest number among all the common factors of the given numbers.
     
  • The HCF (Highest Common Factor) of two natural numbers \(x\) and \(y\) is the largest possible number which divides both \(x\) and \(y\).

From the above example, the common factors of \(18\) and \(27\) are \(1,3\) and \(9\).

Among these numbers, \(9\) is the highest (largest) number.

Highest common factor or HCF definition explained using HCF of 18 and 27

Thus, the HCF of \(18\) and \(27\) is \(9\).

This is written as:

\[\text{HCF (18,27)} = 9 \]

HCF Examples

Using the above definition, the HCF of:

  • \(60\) and \(40\) is \(20\), i.e,
    \[\text { HCF} (60,40) = 20 \]
  • \(100\) and \(150\) is \(50\), i.e,
    \[\text { HCF} (150,50) = 50 \]
  • \(144\) and \(24\) is \(24\), i.e,
    \[\text { HCF} (144,24) = 24 \]
  • \(17\) and \(89\) is \(1\), i.e,
    \[\text { HCF} (17,89) = 1\]

But how to find HCF of two numbers?

Methods of Finding HCF

There are 3 methods to calculate the HCF of two numbers: 

  1. HCF by listing out the common factors
  2. HCF by prime factorization 
  3. HCF by division method

Let us discuss each method in detail.

1. HCF by Listing out the Common Factors

Let us understand this method using the example below.

Example:

What is the HCF of \(30\) and \(42\)?

Solution:

Step 1 - List out the factors of each number.

Factors of \(30\) - \(1, 2, 3, 5, 6, 10, 15, 30\)

Factors of \(42\) - \(1, 2, 3, 6, 7, 14, 21, 42\)

Step 2 - Circle all the common factors.

Finding HCF by listing out the common factors

Step 3 - \(6\) is the common factor and the greatest one.

Hence, the HCF of \(30\) and \(42\) is \(6\).

2. HCF by Prime Factorisation

Let us learn this method using the examples below.

Example 1:

What is the HCF of \(60\) and \(90\)?

Solution:

Step 1 - Represent the numbers in the prime factored form.

Finding HCF by prime factorisation: HCF of 60 and 90Finding HCF by prime factorisation: HCF of 60 and 90

Thus,

\[\begin{align} 60&= 2 \times 2 \times 3 \times 5\\ 90&=2 \times 3\times 3 \times 5 \end{align} \]

Step 2 - HCF is the product of the factors that are common to each of the given numbers.

HCF by prime factorisation: HCF of 60 and 90

This may seem confusing to you. Let us look at the following Venn diagram to understand this method better.

Venn diagram of HCF by prime factorisation: HCF pof 60 and 90

Thus, HCF of \(60\) and \(90\) = \(2 \times 3 \times 5\) = \(30\).

Hence,

HCF of 60 and 90 = 30

Example 2:

What is the HCF of \(120\) and \(144\)?

Solution:

Step 1 - Represent the numbers in the prime factored form.

HCF by prime factorisation: HCF of 120 and 144      HCF by prime factorisation: HCF of 120 and 144

Thus,

\[ \begin{align}
120&=2^{3} \times 3^{1} \times 5^{1}\\
144&=2^{4} \times 3^{2}
\end{align} \]

Step 2 - HCF is the product of the smallest power of each common prime factor:

Thus,

\[ \mathrm{HCF}(120, 144)=2^{3} \times 3^{1} = 24 \]

Hence,

HCF of 120 and 144 = 24

3. HCF by Division Method

Let us learn this method using the example below.

Example:

Find the HCF of \(198\) and \(360\) using the division method.

Solution:

Among the given two numbers, \(360\) is the larger number and \(198\) is the smaller number.

Step 1 - Divide the larger number by the smaller number using long division.

HCF by division method: HCF of 198 and 360

Step 2 - If the remainder is \(0\), then the divisor is the HCF.

If the remainder is NOT \(0\), then make the remainder of the above step as the divisor and the divisor of the above step as the dividend and perform long division again.

HCF by division method: HCF of 198 and 360

Step 3 - If the remainder is \(0\), then the divisor of the last division is the HCF.

If the remainder is NOT \(0\), then we have to repeat step 2 until we get the remainder as \(0\).

HCF by division method: HCF of 198 and 360

Therefore, the HCF of the given two numbers is the divisor of the last division.

In this case, the divisor of the last division is \(18\).

Thus,

HCF of 198 and 360 = 18

HCF of Multiple Numbers

The process of finding the HCF of multiple numbers is the same as explained above for the methods “HCF by listing out the common factors” and “HCF by prime factorisation.”

However, the method to find the HCF of multiple numbers by division method is different.

1. To find the HCF of three numbers:

  • First, we will find the HCF of two of the numbers.
  • Next, we will find the HCF of the third number and the HCF of the first two numbers. 

Example:

Find the HCF of \(126, 162\) and \(180\).

Solution:

First, we will find the HCF of the two numbers \(126\) and \(162\).

HCF of multiple numbers: HCF of 126, 162 and 180

Thus,

\[\text{HCF of 126 and 162}=18  \,\,\,\,\, \rightarrow (1)\]

Next, we will find the HCF of the third number, which is \(180\), and the above HCF \(18\).
HCF of multiple numbers: HCF of 126, 162 and 180
Thus,

\[\text{HCF of 180 and 18}=18  \,\,\,\,\, \rightarrow (2)\]

From (1) and (2),

HCF of \(126, 162\) and \(180\) is \(18\)

2. To find the HCF of four numbers:

  • First, we will find the HCF of the two pairs of numbers separately.
  • Next, we will find the HCF of the HCFs of the two pairs of numbers.

Example:

Find the HCF of \(72,140,256\) and \(24\).

Solution:

First, we will find the HCF of the first two numbers \(72\) and \(140\).

HCF of multiple numbers: HCF of 72, 140, 256 and 24

Thus,

\[\text{HCF of 72 and 140}=4  \,\,\,\,\, \rightarrow (1)\]

Next, we will find the HCF of the last two numbers \(256\) and \(24\).

HCF of multiple numbers: HCF of 72, 140, 256 and 24

Thus,

\[\text{HCF of 256 and 24}=8  \,\,\,\,\, \rightarrow (2)\]

Now, we will find the HCF of the HCFs that are in equation (1) and equation (2), i.e., we will find the HCF of \(4\) and \(8\).

HCF of multiple numbers: HCF of 72, 140, 256 and 24

Thus,

\[\text{HCF of 4 and 8}=4\]

HCF of \(72,140,256\) and \(24\) is \(4\)

HCF of Prime Numbers

We know that a prime number has only two factors, \(1\) and itself.

Let us consider two prime numbers \(2\) and \(7\), and find their HCF by listing their factors.

Factors of \(2\) are \(1, 2\).

Factors of \(7\) are \(1, 7\).

We can see that the only common factor of \(2\) and \(7\) is \(1\) and hence, this is the HCF.

Thus,

HCF of prime numbers = \(1\)

Highest Common Factor (HCF) Calculator

The following "Highest Common Factor Calculator" would show the HCF of the numbers that you enter.

Enter the numbers you wish to find the HCF of and click on the Go button:


Solved Examples

Example 1

 

 

Find the HCF of \(6, 72\) and \(120\) by using the "listing factors" method. 

Solution:

The given numbers are \(6, 72\) and \(120\).

We will find the factors of each of these numbers.

Factors of \(6\) = \(1, 2, 3, 6\)

Factors of \(72\) = \(1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 \)

Factors of \(120\) = \(1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120\)

Now, circle the common factors.

HCF by listing common factors: HCF of 6, 72 and 120

\(6\) is the only common factor and hence, it is the HCF.

HCF of \(6, 72\) and \(120\) = \(6\)
Example 2

 

 

Find the HCF of \(168, 252\) and \(288\) by the "prime factorisation method."

Solution:

The given numbers are \(168, 252\) and \(288\).

We will find the prime factorisation of each of these numbers.

HCF by prime factorisation: HCF of 168, 252 and 288

HCF by prime factorisation: HCF of 168, 252 and 288

HCF by prime factorisation: HCF of 168, 252 and 288

Thus,

\[ \begin{align}168&=2^{3} \times 3^{1} \times 7^{1}\\[0.3cm]252&=2^{2} \times 3^{2} \times 7^{1}\\[0.3cm]288&=2^{5} \times 3^{2}\end{align} \]

The HCF of these three numbers will be the product of the smallest power of each common prime factor of the three numbers.

Thus,

\[\text{HCF of 168, 252 and 288}  = 2^2 \times 3^1 = 12 \]

Hence,

HCF of \(168, 252\) and \(288\) = 12
Example 3

 

 

Find the HCF of \(9000\) and \(980\) using the "division method."

Solution:

Among the given numbers, \(9000\) is the smallest and \(980\) is the largest.

We will divide the larger number by the smaller number.

Next, we will make the remainder as the divisor and the last divisor as the dividend and divide again.

We will repeat this process until the remainder is \(0\).

HCF by division method: HCF of 9000 and 980

Thus,

HCF of \(9000\) and \(980\) = \(20\)

 

 
Thinking out of the box
Think Tank
  1. Two positive numbers \(a\) and \(b\) are written as \(a=x^3y^2\)  & \(b=xy^3\); \(x\) and \(y\) are prime numbers. Find HCF \((a,b)\).
  2. What is the HCF of coprime numbers?

Practice Questions

 
 
 
 
 
 

 


Frequently Asked Questions (FAQs)

1. What is the Highest Common Factor?

  • The HCF (Highest Common Factor) of two or more numbers is the highest number among all the common factors of the given numbers.
     
  • The HCF (Highest Common Factor) of two natural numbers \(x\) and \(y\) is the largest possible number which divides both \(x\) and \(y\).

You can learn the definition of HCF and view the HCF examples under the "HCF (Highest Common Factor) Definition" section of this page.

2. How do you find the HCF in math?

There are 3 methods to calculate the HCF of two numbers: 

  1. HCF by listing out the common factors
  2. HCF by prime factorization 
  3. HCF by division method


These methods are explained in detail with examples under the section titled "Methods of Finding HCF" of this page.

3. What are the properties of HCF?

The properties of HCF are:

  • HCF of two or more numbers divides each of the numbers without a remainder. 
  • HCF of two or more numbers is a factor of each of the numbers.
  • HCF of two or more numbers is always less than or equal to each of the numbers.
  • HCF of two or more prime numbers is \(1\) always.

4. Where can I find the HCF calculator?

We can find the HCF calculator under the "Highest Common Factor Calculator" section of this page. 

You can enter a list of numbers in this calculator and it will show the HCF of the entered numbers.

 
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