How to Solve HCF and LCM Questions with Formula and Examples
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 Practice Questions with Solutions
1. What is HCF and LCM in maths?
The HCF (Highest Common Factor) is the greatest number that divides two or more numbers exactly, while the LCM (Least Common Multiple) is the smallest number that is a multiple of two or more numbers.
- HCF is also called GCD (Greatest Common Divisor).
- LCM is the smallest common multiple shared by the numbers.
- Example: For 12 and 18, HCF = 6 and LCM = 36.
2. How do you find the HCF of two numbers?
You can find the HCF using the prime factorization method or the division method.
- Step 1: Write the prime factors of each number.
- Step 2: Identify common prime factors.
- Step 3: Multiply the common factors with the smallest powers.
3. How do you calculate the LCM of two numbers?
The LCM is calculated by multiplying the highest powers of all prime factors present in the numbers.
- Step 1: Write prime factorization.
- Step 2: Take each prime factor with the highest power.
- Step 3: Multiply them.
4. What is the formula relating HCF and LCM?
The formula connecting HCF and LCM of two numbers is HCF × LCM = Product of the two numbers.
- If two numbers are a and b, then:
- HCF(a, b) × LCM(a, b) = a × b
5. What is the difference between HCF and LCM?
The HCF is the greatest common divisor of numbers, while the LCM is the smallest common multiple.
- HCF divides the numbers exactly.
- LCM is divisible by the given numbers.
- HCF is usually smaller than or equal to the numbers.
- LCM is greater than or equal to the numbers.
6. How do you find HCF using the division method?
The division method finds the HCF by repeatedly dividing until the remainder becomes zero.
- Step 1: Divide the larger number by the smaller number.
- Step 2: Replace the larger number with the divisor and the smaller number with the remainder.
- Step 3: Repeat until remainder = 0.
7. Can you give an example of an HCF and LCM word problem?
HCF and LCM word problems often involve grouping or repeated events. Example: Two bells ring every 6 and 8 minutes. When will they ring together again?
- Find LCM of 6 and 8.
- 6 = 2×3, 8 = 2³.
- LCM = 2³×3 = 24 minutes.
8. What is the HCF and LCM of 12 and 15?
The HCF of 12 and 15 is 3 and the LCM is 60.
- 12 = 2²×3
- 15 = 3×5
- HCF = 3
- LCM = 2²×3×5 = 60
9. What are the properties of HCF and LCM?
The key properties of HCF and LCM help simplify calculations.
- HCF of co-prime numbers is 1.
- LCM of co-prime numbers is their product.
- HCF × LCM = Product of numbers (for two numbers).
- LCM is always divisible by each given number.
10. What are common mistakes in HCF and LCM questions?
Common mistakes in HCF and LCM questions include mixing up smallest and greatest values or using incorrect prime factors.
- Confusing HCF with LCM.
- Not taking highest powers for LCM.
- Not taking lowest powers for HCF.
- Arithmetic errors in multiplication.
