Understanding HCF: What It Is and How to Find It [Step-by-Step Guide]

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What is HCF?

The term common factor, refers to a factor that two or more numbers share. The largest number that divides both x and y is known as the HCF (Highest Common Factor) of two natural numbers, x and y. There is no remainder when this biggest factor is divided by any of the numbers given. 

HCF can be assessed using two or more numbers. Greater Common Measure(GCM) and Greatest Common Divisor(GCD) are further names for HCF.

Let’s understand HCF with the help of an example:

Example: Find the HCF of 60 and 24.

Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 

Factor of 60

Factors of 24 = 1, 2, 3, 4, 6, 8, 12 and 24 

Factor of 24

Common factors of 24 and 60 = 1, 2, 3, 4, 6, 12

Highest Common Factor = 12

HCF Full Form?

HCF stands for Highest Common Factor. The highest common factor is derived by identifying all common factors between two or more numbers.

How to Find HCF of Two Numbers

We can calculate the HCF of any given number using one of two methods: 

1. HCF by Division Method

2. HCF by Prime Factorization Method

Let’s talk about each of these two approaches separately in this blog.

HCF by Division Method

Follow the steps below to find the HCF of numbers using the division method.

Step 1: The first step is to divide the larger by the smaller number. 

Step 2: Repeat the long division, this time making the remainder of the preceding step the divisor, and the divisor of the preceding step the dividend. 

Step 3: Long division should be continued until the remainder equals zero. 

Step 4: When the remainder is zero, HCF is the final divisor.

To understand the division method better, let’s find the HCF of 16 and 52

HCF by Division Method

HCF by Prime Factorisation Method

The methods below will show you how to calculate a number’s HCF using the prime factorization method. 

Step 1: Each of the given numbers should be divided into prime factors. 

Step 2: List each of the numbers’ common prime factors. 

Step 3: The sum of the prime factors for the given numbers is the HCF. 

To understand the prime factorization method better, let’s find the HCF of 30 and 45. 

Prime factorization of 30 = 2 x 3 x 5             

Prime factorization of 45 = 3 x 3 x 5

Common prime factors of 30 and 45 = 3 and 5, 

Thus Highest Common Factor of 30 and 45 = 15 (3 x 5)

HCF Questions for Practice

It is crucial for students to have a solid understanding of HCF in order to pick up more complex concepts rapidly. In number theory and other areas like factorization, permutations, combinations, etc. they are mostly used.

In real-life situations we use HCF, when we wish to divide something into smaller portions, organize something into rows or groups, give more stuff to large groups, determine how many people we need to invite for a party and much more.

Find the HCF of the following Numbers

Prime Factorisation Method

Prime factorisation of 24 = 2 x 2 x 2 x 3

Prime factorisation of  of 36 = 2 x 2 x 3 x 3

Common prime factors = 2, 2, 3

Highest Common Factor of 24 and 36 = 2 x 2 x 3 = 12

Prime Factorisation Method

Prime factorisation of 92 = 2 x 2 x 23

Prime factorisation of 152 = 2 x 2 x 2 x 19

Common prime factors = 2, 2

Highest Common Factor of 92 and 152 = 2 x 2 = 4

Prime Factorisation Method

Prime factorisation of 8 = 2 × 2 × 2

Prime factorisation of 9 = 3 × 3

Prime factorisation of 25 = 5 × 5

Hence, the Highest Common Factor of 8, 9 and 25 = 1 (because there is no common factor)

Prime Factorisation Method

Prime factorisation of 12 = 2 x 2 x 3 

Prime factorisation of 18 = 2 x 3 x 3

Common prime factors = 2, 3

Highest Common Factor of 12 and 18 = 2 x 3 = 6

Prime Factorization Method

Prime factorization of 196 = 2 × 2 × 7 × 7 

Prime factorization of 38220 = 2 × 2 × 3 × 5 × 7 × 7 × 13

Common prime factors = 2, 2, 7, 7 

Therefore Highest Common Factor of 196 and 38220 = 2 × 2 × 7 × 7 = 196

Prime Factorisation Method

Prime factorisation of 36 = 2 × 2 × 3 × 3

Prime factorisation of 84 = 2 × 2 × 3 × 7

Common prime factors = 2, 2, 3

Highest Common Factor of 36 and 84 = 2 × 2 × 3 = 12

Prime Factorisation Method

Prime factorisation of = 3 × 3 × 3 × 5

Prime factorisation of = 3 × 3 × 5 × 5

Common prime factors = 3, 3, 5

Highest Common Factor of 135 and 225 = 3 x 3 x 5 = 45

 

Prime Factorisation Method

Prime factorisation of 96 = 2 × 2 × 2 × 2 × 2 × 3

Prime factorisation of = 2 × 2 × 101 

Common prime factors = 2, 2

Highest Common Factor of 96 and 404 = 2 x 2 = 4

Prime Factorisation Method

Prime factorisation of 18 = 2 × 3 × 3

Prime factorisation of 54 = 2 × 3 × 3 × 3

Prime factorisation of 81 = 3 × 3 × 3 × 3

Common prime factors = 3, 3

Highest Common Factor of 18, 54 and 81 = 3 x 3 = 9

Prime Factorisation Method

Prime factorisation of 18 = 2 × 3 × 3

Prime factorisation of 24 = 2 x 2 x 2 x 3

Common prime factors = 2, 3

Highest Common Factor of 18 and 24 = 2 x 3 = 6

Prime Factorisation Method

Prime factorisation of 26 = 2 × 13

Prime factorisation of 91 = 7 × 13

Common prime factor = 13

Highest Common Factor of 26 and 91 = 13

Prime Factorisation Method

Prime factorisation of 510 = 2 × 3 × 5 × 17

Prime factorisation of 92 = 2 × 2 × 23

Common prime factor = 2

Highest Common Factor of 510 and 92  = 2

Prime Factorisation Method

Prime factorisation of 70 = 2 × 5 × 7

Prime factorisation of 105 = 3 × 5 × 7

Prime factorisation of 175 = 5 × 5 × 7

Common prime factors = 5 and 7

Highest Common Factor of 70, 105 and 175= 5 x 7 = 35

Prime Factorisation Method

Prime factorisation of 12 = 2 × 2 × 3 

Prime factorisation of 45 = 3 × 3 × 5

Prime factorisation of  75 = 3 × 5 × 5

Common prime factor = 3

Highest Common Factor of 12, 45 and 75 = 3

 

Prime Factorisation Method

Prime factorisation of 144 = 2 × 2 × 2 × 2 × 3 × 3

Prime factorisation of 198 = 2 × 3 × 3 × 11

Common prime factors = 2, 3 and 3

Highest Common Factor of 144 and 198 = 2 x 3 x 3 = 18

 

Prime Factorisation Method

Prime factorisation of 25 = 5 × 5 

Prime factorisation of 40 = 2 × 2 × 2 × 5

Common prime factor = 5

Highest Common Factor of 25 and 40 = 5

Prime Factorisation Method

Prime factorisation of 336 = 2 × 2 × 2 × 2 × 3 × 7 

Prime factorisation of 54 = 2 × 3 × 3 × 3

Common prime factors = 2, 3

Highest Common Factor of 336 and 54 = 2 x 3 = 6

 

Prime Factorisation Method

Prime factorisation of 34 = 2 × 17 

Prime factorisation of 102 = 2 × 3 × 17

Common prime factors = 2 and 17

Highest Common Factor of 34 and 102 = 2 x 17 = 34

 

Prime Factorisation Method

Prime factorisation of 4052 = 2 × 2 × 1013

Prime factorisation of 12576 = 2 × 2 × 2 × 2 × 2 × 3 × 131

Common prime factors = 2 and 2

Highest Common Factor of 4052 and 12576 = 2 x 2 = 4

 

 

Prime Factorisation Method

Prime factorisation of 91 = 7 × 13 

Prime factorisation of 112 = 2 × 2 × 2 × 2 × 7

Prime factorisation of 49 = 7 × 7

Common prime factor = 7

Highest Common Factor of 91, 112 and 49 = 7

HCF of Two Co-Prime Numbers

When the highest common factor (HCF) between two numbers is 1, that pair of numbers is referred to as co-prime. 

For example, the only common factor of 8 and 45 is 1.

Prime factorisation of 8 = 2 x 2 x 2

Prime factorisation of 45 = 5 x 3 x 3

Common prime factor of 8 and 45  = 1

Highest Common Factor of 8 and 45 = 1

Therefore, 8 and 45 are co-prime, and their HCF is 1.

Conclusion

Planning, estimating, and dividing things can be made much easier by having a basic understanding of HCF. The real-world implementations of these ideas help youngsters become more adept at problem-solving and critical thinking. Thanks to igebra.ai, children may quickly comprehend a range of mathematical ideas and how they relate to everyday life. To support children’s visual learning and problem-solving mindset, it offers a variety of educational tools and interactive activities.