How to Find HCF Easily: Step-by-Step Methods with Examples
The concept of HCF (Highest Common Factor) plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. It helps students simplify problems, find greatest divisors, and solve a wide range of questions in school and competitive exams.
What Is HCF (Highest Common Factor)?
The Highest Common Factor (HCF) of two or more numbers is the largest number that divides each of them exactly, leaving no remainder. In other words, it is the greatest number that is a factor of all the given numbers. You’ll find this concept applied in topics such as LCM (Least Common Multiple), Number System, and Factors.
Key Formula for HCF
Here’s the handy formula and shortcut for finding HCF using the product of numbers and their LCM:
\( \text{HCF} \times \text{LCM} = \text{Product of the Numbers} \)
Or, for two numbers A and B:
\( \text{HCF}(A, B) = \frac{A \times B}{\text{LCM}(A, B)} \ )
Cross-Disciplinary Usage
HCF is not only useful in Maths but also plays an important role in Physics (like oscillations and waves), Computer Science (like cryptography and algorithms), and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions, especially in simplifying ratios, fractions, and problem-solving.
Step-by-Step Illustration: How to Find HCF
- **By Listing Factors:
Example: Find HCF of 12 and 18. - **By Prime Factorization:**
Example: HCF of 36 and 48.
36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3 - **By Division (Euclidean method):**
Example: HCF of 20 and 32.
2. List the factors of 18: 1, 2, 3, 6, 9, 18
3. Find common factors: 1, 2, 3, 6
4. Highest one = 6.
2. HCF = 2 × 2 × 3 = 12
2. Divide 20 by 12 ⇒ remainder 8
3. Divide 12 by 8 ⇒ remainder 4
4. Divide 8 by 4 ⇒ remainder 0.
5. Last divisor is 4
Speed Trick or Vedic Shortcut
Here’s a quick shortcut that helps solve problems faster when working with HCF. Many students use this trick during timed exams to save crucial seconds.
Example Trick: For two numbers, subtract the smaller number from the larger one, and keep repeating with the new pair (bigger, difference) till you get 0. The last non-zero number is your HCF. This is the essence of the Euclidean algorithm.
- HCF of 45 and 75:
75 – 45 = 30
HCF(45, 30): 45 – 30 = 15
HCF(30, 15): 30 – 15 = 15
HCF(15, 15): 15 – 15 = 0
So, HCF is 15
Tricks like this aren’t just cool — they’re practical in competitive exams like NTSE, Olympiads, and even JEE. Vedantu’s live sessions include more such shortcuts to build your calculation speed.
Try These Yourself
- Find the HCF of 24 and 36 by three different methods.
- What is the HCF of 16, 24, and 32?
- If HCF of two numbers is 8 and their LCM is 48, what can the numbers be?
- Which is greater — LCM or HCF — for any two numbers?
Frequent Errors and Misunderstandings
- Mixing up HCF and LCM — HCF is always the greatest divisor, not the multiple.
- Missing out common factors when using the prime factorization method.
- Forgetting to stop at remainder 0 in the long division method.
Relation to Other Concepts
The idea of HCF connects closely with topics such as LCM (Least Common Multiple) and Prime Factorization. Mastering HCF helps with understanding more advanced number theory, simplifying fractions, and solving word-based real-life problems.
Classroom Tip
A quick way to remember HCF: “HCF = Highest Common Factor = Greatest Divisor Common to All.” Draw factor trees or use Venn diagrams to visualize overlapping factors. Vedantu’s teachers often use colorful charts and time-saving tricks during live classes to help students master HCF concepts quickly and confidently.
Wrapping It All Up
We explored HCF (Highest Common Factor)—from its simple definition, standard formulas, step-by-step calculations, shortcut tricks, and common mistakes, to its strong connection with LCM and other number system concepts. Continue practicing HCF problems and explore interactive resources on Vedantu to become completely confident in solving any exam or real-life question involving HCF.
More Helpful Links
FAQs on HCF (Highest Common Factor) in Maths: Meaning, Methods & Practice
1. What is HCF in Maths?
The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides each of two or more integers without leaving a remainder. It represents the largest common factor among the given numbers. For example, the HCF of 12 and 18 is 6.
2. How do you find the HCF of two or more numbers?
There are three primary methods to find the HCF:
• Listing Factors: List all factors of each number and identify the largest common factor.
• Prime Factorization: Find the prime factorization of each number. The HCF is the product of the common prime factors raised to the lowest power.
• Division Method (Euclidean Algorithm): Continuously divide the larger number by the smaller number until the remainder is 0. The last non-zero remainder is the HCF.
3. What is the difference between HCF and LCM?
The Highest Common Factor (HCF) is the largest number that divides two or more integers without a remainder, while the Least Common Multiple (LCM) is the smallest positive integer that is a multiple of two or more integers. They are inversely related; a larger HCF implies a smaller LCM, and vice versa. For example, for numbers 6 and 8, HCF is 2 and LCM is 24.
4. What is the full form of HCF?
HCF stands for Highest Common Factor. It is also known by its synonym, Greatest Common Divisor (GCD).
5. What are the three common methods for calculating HCF?
The three common methods are: Listing Factors, Prime Factorization, and the Division Method (Euclidean Algorithm).
6. Can you find HCF using a calculator or online tool?
Yes, many online calculators and software programs can compute the HCF of numbers. Vedantu also offers such tools to assist with calculations.
7. Why is HCF important in real-life problems and exams?
HCF finds applications in various real-world scenarios such as dividing quantities equally (e.g., sharing sweets among friends), simplifying fractions, and solving problems involving measurements. It also forms a crucial part of many mathematical problems in school and competitive exams.
8. How do you find the HCF of more than two numbers (e.g., 184, 230, 276)?
The methods for finding the HCF extend to more than two numbers. For the division method, find the HCF of any two numbers, then find the HCF of that result and the next number, and so on. Prime factorization remains the same; find the prime factors of each number and identify the common factors raised to the lowest power.
9. What is the fastest method for HCF in competitive exams?
The division method (Euclidean Algorithm) is generally considered the fastest for larger numbers in competitive exams due to its efficiency. However, for smaller numbers, prime factorization might be quicker if you can readily identify the factors.
10. How does prime factorization differ from the division method in finding HCF?
Prime factorization breaks down each number into its prime factors, identifying the common ones to calculate the HCF. The division method uses successive divisions to find the greatest common divisor. Prime factorization is conceptually simpler but can be slower for very large numbers, whereas the division method is more efficient for larger numbers but may be less intuitive.
11. Can HCF be used for fractions or decimals?
While HCF is primarily defined for integers, it can be extended to fractions and decimals by first converting them to equivalent fractions with a common denominator and then finding the HCF of the numerators. For decimals, convert them to fractions first.
12. How to avoid common mistakes while finding HCF in word problems?
Carefully identify the relevant quantities that need to be considered for calculating the HCF. Ensure you understand the problem's context before applying the chosen method. Double-check your calculations to avoid arithmetic errors. Practice diverse problems to enhance your problem-solving skills and build confidence.
