LCM and HCF Solved Problems (With Step-by-Step Explanation)
LCM (Least Common Multiple) and HCF (Highest Common Factor) are two of the most important concepts in arithmetic. These topics form the foundation for higher mathematics and are frequently asked in school exams, Olympiads, and competitive exams. Understanding LCM and HCF helps students solve problems related to time, work, grouping, and number patterns.
In this article, you will find fully solved LCM and HCF problems with clear explanations. Each problem is solved step by step so that students can understand not just the answer, but also the method used.
Important Concepts and Formulas
HCF (Highest Common Factor): The greatest number that divides two or more given numbers exactly.
LCM (Least Common Multiple): The smallest number that is exactly divisible by two or more given numbers.
Key Relationship:
For any two positive numbers:
LCM × HCF = Product of the two numbers
Solved Problems
Problem 1: Find the HCF of 24 and 36
Solution:
Step 1: Write the prime factors.
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
Step 2: Select common prime factors with the smallest powers.
Common factors = 2 × 2 × 3
HCF = 12
Problem 2: Find the LCM of 15 and 20
Solution:
15 = 3 × 5
20 = 2 × 2 × 5
Step 1: Take the highest power of each prime factor.
LCM = 2² × 3 × 5
LCM = 60
Problem 3: Find the HCF and LCM of 18 and 27
Solution:
18 = 2 × 3 × 3
27 = 3 × 3 × 3
HCF = 3 × 3 = 9
LCM = 2 × 3 × 3 × 3 = 54
Problem 4: Find the smallest number divisible by 8, 12, and 15
Solution:
8 = 2³
12 = 2² × 3
15 = 3 × 5
LCM = 2³ × 3 × 5
Required number = 120
Problem 5: The HCF of two numbers is 6 and their product is 180. Find the LCM.
Solution:
Using the formula:
LCM × HCF = Product of numbers
LCM × 6 = 180
LCM = 30
Common Mistakes Students Make
- Confusing LCM with HCF
- Missing prime factors during factorization
- Using the LCM × HCF formula incorrectly
- Applying the formula to more than two numbers
Practice Tips
✔ Always write prime factorizations neatly
✔ Double-check common and uncommon factors
✔ Use the LCM & HCF calculator to verify answers
👉 Related Learning Tools: