Table of Contents 

We at Cuemath believe that Math is a life skill. Our Math Experts focus on the “Why” behind the “What.” Students can explore from a huge range of interactive worksheets, visuals, simulations, practice tests, and more to understand a concept in depth.

Book a FREE trial class today! and experience Cuemath’s LIVE Online Class with your child.

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:

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 \(18\).

---

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.

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

---

• 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.

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?

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.

CLUEless in Math? Check out how CUEMATH Teachers will explain Highest Common Factor to your kid using interactive simulations & worksheets so they never have to memorise anything in Math again!

Explore Cuemath Live, Interactive & Personalised Online Classes to make your kid a Math Expert. Book a FREE trial class today!

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 \(162\).

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 \(18\).

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 \(140\).

Thus,

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

Next, we will find the HCF of the last two numbers \(256\) 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 \(8\).

Thus,

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

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

---

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 = \(1\)