Step-by-Step With Highest Common Factor (HCF)

When working with numbers, we often need to simplify problems, especially those involving fractions, ratios, and divisibility. One of the most useful concepts is the Highest Common Factor (HCF).

The HCF helps us find the largest number that divides two or more given numbers without leaving a remainder. It’s also known as the Greatest Common Divisor (GCD).

Also Read: Lowest Common Multiple (LCM)

In this post, we’ll explore some of the guides to finding HCF to includes:

• What HCF means,
• Different methods to find it,
• Worked-out examples,
• Real-life applications,
• Practice questions for students.

What is the Highest Common Factor (HCF)?

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides all of them exactly.

It is also called the Greatest Common Divisor (GCD).

For example:

Find the HCF of 12 and 18.

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18

The highest common factor is 6.

So, HCF of (12, 18) = 6.

Methods of Finding HCF

There are multiple methods to find the Highest Common Factor:

1. Listing Factors Method

We list out all factors of each number and identify the highest common one.

For example: Find HCF of 15 and 25.

• Factors of 15 = 1, 3, 5, 15
• Factors of 25 = 1, 5, 25

Common factors = 1, 5

HCF = 5

So, HCF of (15, 25) = 5.

2. Prime Factorization Method

We break down each number into prime factors, then multiply the common primes with the smallest powers.

For Example: Find HCF of 48 and 60.

• Prime factorization of 48 = 2⁴ × 3
• Prime factorization of 60 = 2² × 3 × 5

Take common factors with the lowest powers:

• 2²×3=12.

So, HCF of (48, 60) = 12.

3. Division (Euclidean Algorithm)

This method uses division repeatedly until the remainder becomes 0. The divisor at that stage is the HCF.

For Example: Find HCF of 56 and 72.

Step 1: Divide 72 by 56 → remainder = 16.

Step 2: Divide 56 by 16 → remainder = 8.

Step 3: Divide 16 by 8 → remainder = 0.

HCF = 8.

4. Short Division Method

Similar to LCM’s division method, but here we take only the common prime divisors.

For Example: Find HCF of 36, 60, and 84.

Step   36      60      84

÷2      18      30      42

÷2      9        15      21

÷3      3        5        7

÷…    –        –        –

Take only the divisors common to all: 2 × 3 = 6.

So, HCF of (36, 60, 84) = 6.

More Worked-Out Examples

Example 1:

Find HCF of 20 and 28.

Factors of 20 = 1, 2, 4, 5, 10, 20

Factors of 28 = 1, 2, 4, 7, 14, 28

Common = 1, 2, 4

HCF = 4

Example 2:

Find HCF of 81 and 108 using prime factorization.

81 = 3⁴

108 = 2² × 3³

Common = 3³ =27.

HCF of (81, 108) = 27

Example 3: Real-Life Application

A farmer has 60 mangoes and 48 oranges. He wants to pack them in equal groups without mixing fruits. What is the largest number of fruits in each group?

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Find HCF of (60, 48).

60 = 2² × 3 × 5

48 = 2⁴ × 3

Common = 2²×3=12.

Each group will have 12 fruits.

Practice Problems

Try solving these:

1. Find HCF of 16 and 24.
2. Find HCF of 72, 120, and 144.
3. Find HCF of 25 and 45.
4. Two ropes of length 120 cm and 150 cm need to be cut into equal pieces with no remainder. What is the maximum length of each piece?

Conclusion

The Highest Common Factor (HCF) is an important concept in mathematics that helps in simplifying numbers, solving fraction problems, and dealing with real-life applications.

We explored:

• Different methods (listing, prime factorization, Euclidean algorithm, and short division),
• Worked examples,
• Real-world word problems,
• Practice exercises.

By mastering HCF, you’ll strengthen your foundation in arithmetic, fractions, and number theory.