How to Solve HCF and LCM Questions Easily for Exams
The concept of HCF and LCM questions is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Knowing how to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) is important for students of all classes, for board exams, and for competitive exams.
Understanding HCF and LCM Questions
HCF and LCM questions ask you to determine the largest number that divides given numbers without leaving a remainder (HCF), and the smallest number that is a multiple of two or more numbers (LCM). These concepts are widely used in solving arithmetic problems, time scheduling, and understanding factors and multiples. You may encounter them in school-level worksheets, entrance tests, and even in everyday problem-solving.
Methods to Find HCF and LCM
To solve HCF and LCM questions, you can use different methods, each with step-by-step processes. Here are the three most common methods:
| Method | HCF | LCM |
|---|---|---|
| Prime Factorization | Take common prime factors and multiply them | Multiply all prime factors, using common ones only once |
| Division Method | Successively divide numbers by their common factors | Multiply numbers, then divide by HCF |
| Listing Method | List all factors and find the greatest common one | List multiples and find the smallest common one |
Each method makes it easier to solve HCF and LCM questions for all types of numbers—small or large.
Worked Examples – Solving HCF and LCM Problems
Let’s see step-by-step solutions for common HCF and LCM questions:
Example 1: Find HCF of 24 and 36 (Listing Method)
1. List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
2. List the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
3. Identify the highest common factor: 12
Final Answer: HCF = 12
Example 2: Find LCM of 3 and 4 (Listing Multiples)
1. Multiples of 3: 3, 6, 9, 12, 15, 18...
2. Multiples of 4: 4, 8, 12, 16, 20...
3. The smallest common multiple: 12
Final Answer: LCM = 12
Example 3: Find HCF of 135 and 225 (Prime Factorization)
1. Prime factors of 135: 3 × 3 × 3 × 5
2. Prime factors of 225: 3 × 3 × 5 × 5
3. Common prime factors: 3 × 3 × 5 = 45
Final Answer: HCF = 45
Classwise HCF and LCM Questions
Here are some sample HCF and LCM questions for different classes:
Class 5
1. Find the HCF and LCM of 8 and 12.
2. Two bells ring every 6 and 8 minutes. When will they ring together next?
Class 6
1. What is the LCM of 15, 20, and 30?
2. Find the HCF of 60 and 90 by prime factorization.
Class 7
1. If the HCF of two numbers is 13 and their product is 2028, find their LCM.
2. What is the LCM of 18, 24, and 36?
Class 10
1. Show that LCM(a, b) × HCF(a, b) = a × b for any two positive integers.
2. Find the HCF and LCM of 420 and 130 by division method.
Competitive Exam Level Questions
HCF and LCM questions are common in competitive exams like SSC-CGL, banking, and entrance tests. Here are examples with answers:
| Question | Answer |
|---|---|
| The HCF of 48 and 180 by prime factorization? | 12 |
| What is the LCM of 24, 36, and 60? | 360 |
| Find the smallest number divisible by 12, 15, and 20. | 60 |
Practice Worksheet and PDF
To practice more HCF and LCM questions with answers, download free printable worksheets from Vedantu. PDF resources are available to help with offline study and fast revision.
Tips and Tricks for HCF and LCM Questions
- Always write out prime factors for clarity before choosing HCF or LCM.
- For two numbers: HCF × LCM = product of the numbers.
- If numbers are co-prime, the HCF is 1 and LCM is their product.
- Use mental math shortcuts to list out multiples/factors for smaller numbers.
- Double-check division steps in the division method to avoid mistakes.
Common Mistakes to Avoid
- Confusing HCF (the greatest factor) with LCM (the smallest multiple).
- Missing common factors in prime factorization.
- Not reducing fractions to lowest terms before solving.
Real-World Applications
The concept of HCF and LCM questions appears in areas such as synchronizing event cycles, dividing things evenly, packaging, designing timetables, and even in banking systems. Learning these skills with Vedantu shows maths is useful far beyond exams!
We explored the idea of HCF and LCM questions, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts.
Related Maths Topics on Vedantu
FAQs on HCF and LCM Questions with Stepwise Solutions and Answers
1. How to do HCF and LCM questions?
To solve HCF (Highest Common Factor) and LCM (Lowest Common Multiple) questions, follow these basic steps:
- Prime Factorization: Express each number as a product of primes.
- HCF: Choose the lowest power of each common prime factor.
- LCM: Select the highest power of all prime factors present in any number.
Prime factorization:
- 12 = $2^2 \times 3^1$
- 18 = $2^1 \times 3^2$
LCM = $2^{\max(2,1)} \times 3^{\max(1,2)} = 2^2 \times 3^2 = 36$
To master HCF and LCM questions, practice with various examples and utilize Vedantu’s interactive resources for step-by-step explanations and expert guidance.
2. What is an example of LCM and HCF?
Example using numbers 15 and 20:
- Prime factors of 15: $3 \times 5$
- Prime factors of 20: $2^2 \times 5$
LCM: $2^2 \times 3 \times 5 = 60$
Therefore,
- HCF of 15 and 20 = 5
- LCM of 15 and 20 = 60
3. What is the HCF of 120 and 168?
To find the HCF of 120 and 168:
Prime factorization:
- 120 = $2^3 \times 3^1 \times 5^1$
- 168 = $2^3 \times 3^1 \times 7^1$
So, HCF = $2^3 \times 3^1 = 8 \times 3 = 24$
The HCF of 120 and 168 is 24.
Students can practice more such problems with Vedantu’s expertly curated HCF question banks and video solutions.
4. What is the LCM of 8 9 and 25 and HCF?
To find the LCM and HCF of 8, 9, and 25:
- 8 = $2^3$
- 9 = $3^2$
- 25 = $5^2$
LCM = $2^3 \times 3^2 \times 5^2 = 8 \times 9 \times 25 = 1800$
For HCF: No prime is common in all three numbers, so HCF = 1.
The LCM is 1800 and the HCF is 1.
Dive into similar complex questions with guided explanations using Vedantu’s online learning modules.
5. What are the main differences between HCF and LCM problems?
HCF (Highest Common Factor) and LCM (Lowest Common Multiple) differ in purpose and application:
- HCF identifies the largest number that divides two or more numbers without remainder. It is used in problems requiring division into smaller, identical groups.
- LCM finds the smallest number that is a multiple of two or more numbers. LCM is useful for synchronizing events or finding common cycles.
6. How do you solve HCF and LCM word problems in competitive exams?
To solve HCF and LCM word problems:
- Read the question carefully to understand what is given and what is required.
- Translate the scenario into a mathematical expression involving HCF or LCM.
- Apply prime factorization or division methods, as appropriate.
7. Why are LCM and HCF important in real-life applications?
LCM and HCF are fundamental in various real-life situations such as:
- Scheduling events that repeat after certain periods (using LCM)
- Dividing things equally, like distributing sweets into identical packets (using HCF)
- Solving problems involving gears, traffic lights, and time intervals
8. What are the different methods to find HCF and LCM?
The main methods to calculate HCF and LCM include:
- Prime Factorization Method: Write each number as a product of primes.
- Division Method (for HCF): Divide the larger number by the smaller number and repeat with remainder.
- Listing Multiples or Factors (for small numbers): List all multiples or factors and find the required one.
9. How can students practice HCF and LCM questions effectively?
Students can improve their HCF and LCM skills by:
- Solving a variety of practice problems in Vedantu’s learning platform
- Watching live classes and video tutorials for concept clarity
- Attempting mock tests and worksheets regularly
- Seeking one-on-one doubt clarification with Vedantu’s experienced tutors
