How to Solve HCF and LCM Problems Step by Step
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: Practice with Answers and Solutions
1. What is HCF and LCM in mathematics?
In mathematics, HCF (Highest Common Factor) of two or more numbers is the greatest number that divides them completely. LCM (Lowest Common Multiple) is the smallest multiple common to the given numbers. Both concepts are fundamental in arithmetic and help solve problems involving factors and multiples.
2. How do you find HCF and LCM of two numbers?
To find the HCF and LCM of two numbers, you can use these methods:
- Prime Factorization: Break each number into prime factors, multiply the common prime factors for HCF, and multiply all prime factors (taking highest occurrences) for LCM.
- Division Method: Repeatedly divide by common prime factors until no further division is possible.
- Listing Method: List factors or multiples and identify the highest common factor or lowest common multiple respectively.
3. What are some examples of HCF and LCM questions for class 10?
Examples for class 10 include:
• Find the HCF of 544 and 374 using prime factorization.
• Calculate the LCM of 54 and 60.
• Solve word problems involving real-life applications of HCF and LCM.
These practice problems help build proficiency for board exams and competitive tests.
4. How is the relationship between HCF and LCM used in exams?
The relationship HCF × LCM = Product of the numbers is crucial in exams. It simplifies calculations by allowing you to find one value when the other two are known. This formula is often used to solve tricky numerical problems and verify answers efficiently.
5. Where can I download HCF and LCM questions with answers in PDF?
You can download comprehensive HCF and LCM question PDFs with solutions from trusted educational platforms like Vedantu and BYJU’S. These resources support offline practice and include varied difficulty levels for classes 4 to 10 and competitive exams. Look for official links labeled as ‘hcf and lcm questions pdf’ on such sites.
6. What is the difference between HCF and LCM?
HCF (Highest Common Factor) is the greatest factor common to two or more numbers, while LCM (Lowest Common Multiple) is the smallest multiple common to them. HCF helps in simplifying fractions and solving divisibility problems, whereas LCM is used for finding intervals in time schedules, event synchronizations, and adding fractions with different denominators.
7. Why are students often confused between HCF and LCM methods?
Students often confuse HCF and LCM methods because both involve prime factorization and finding common numbers. The key difference is that for HCF, we take the lowest powers of common prime factors, whereas for LCM, we take the highest powers of all prime factors. Understanding this distinction solves most confusion.
8. Why is HCF not always a factor of the LCM?
HCFLCM is their smallest common multiple. Since LCM may include prime factors not present in the HCF, the HCF is not necessarily a factor of the LCM itself. The important point is that HCF × LCM equals the product of the numbers, linking them indirectly.
9. Why are HCF and LCM important for non-math competitive exams?
HCF and LCM questions test arithmetic aptitude and logical reasoning skills, crucial for many competitive exams like SSC-CGL, banking, and entrance tests. They assess speed, accuracy, and problem-solving ability with numbers, which are foundational for higher quantitative reasoning and exam success.
10. Why do some word problems require both HCF and LCM in the solution?
Certain word problems, especially involving event synchronization or division of resources, need both HCF and LCM.
• HCF is used to find the greatest size or number for equal distribution.
• LCM helps determine when events occur together again.
These dual uses often come up in scheduling, grouping, or partition problems in exams.
11. Why are HCF and LCM questions asked in almost all school exams?
HCF and LCM are fundamental concepts in number theory and arithmetic. They form the base for many mathematical operations such as fraction simplification, ratio problems, and problem-solving skills. Due to their broad applicability and importance in logical thinking, schools include them consistently across grades and exam boards.
