LCM & HCF
SSC GD Exam - Quantitative Aptitude
1. What is LCM?
LCM stands for Least Common Multiple. It is the smallest number that is exactly divisible by two or more given numbers.
Simple Definition:
The LCM of two or more numbers is the lowest common
multiple of those numbers.
Example: Find LCM of 4 and 6
Multiples of 4 → 4, 8, 12, 16, 20, 24, …
Multiples of 6 → 6, 12, 18, 24, 30, …
Common multiples → 12, 24, 36, …
✅ LCM = 12
2. What is HCF?
HCF stands for Highest Common Factor, also known as GCD (Greatest Common Divisor). It is the largest number that exactly divides two or more given numbers.
Simple Definition:
The HCF of two or more numbers is the greatest number that
divides all of them without leaving a remainder.
Example: Find HCF of 12 and 18
Factors of 12 → 1, 2, 3, 4, 6, 12
Factors of 18 → 1, 2, 3, 6, 9, 18
Common factors → 1, 2, 3, 6
✅ HCF = 6
3. Relationship Between LCM and HCF
Important Formula
HCF × LCM = a × b
For any two numbers a and b
Example Verification
For 12 and 18: HCF = 6, LCM = 36
→ 6 × 36 = 216 = 12 × 18 ✅
4. Methods to Find LCM & HCF
| Method | Description | Usefulness |
|---|---|---|
| Prime Factorization | Express numbers as product of prime factors | Useful for small numbers |
| Division Method | Divide by common prime factors until no division is possible | Fast for large numbers |
| Common Multiple/Factor Method | List all multiples or factors to find the least/greatest common | Best for simple understanding |
5. Example Problems
Example 1 (LCM): Find LCM of 8, 12, and 16
→ 8 = 2³
→ 12 = 2² × 3
→ 16 = 2⁴
Take highest powers of all primes → 2⁴ × 3 = 16 × 3 = 48
✅ LCM = 48
Example 2 (HCF): Find HCF of 24, 36, and 60
→ 24 = 2³ × 3
→ 36 = 2² × 3²
→ 60 = 2² × 3 × 5
Take common prime factors with smallest powers → 2² × 3 = 12
✅ HCF = 12
6. Difference Between LCM and HCF
| Basis | LCM | HCF |
|---|---|---|
| Full Form | Least Common Multiple | Highest Common Factor |
| Definition | Smallest number divisible by all | Greatest number dividing all |
| Value | Greater than or equal to each number | Smaller than or equal to each number |
| Relation | LCM × HCF = Product of numbers | Same relation applies |
| Example | LCM(4, 6) = 12 | HCF(4, 6) = 2 |
7. Real-Life Analogy
LCM Example: Traffic Lights
Two traffic lights blink every 12 and 15 seconds.
Both will blink together after:
LCM(12,15) = 60 seconds
→ LCM helps find when events coincide
HCF Example: Apple Distribution
You have 18 and 24 apples to distribute equally.
The maximum number of equal groups:
HCF(18, 24) = 6
→ HCF helps find equal distribution
8. Common Exam Facts (Delhi Police / SSC)
LCM × HCF = Product of Numbers (for two numbers)
LCM ≥ Greatest number
HCF ≤ Smallest number
HCF of co-prime numbers = 1
LCM of co-prime numbers = Product of the numbers
9. Quick Recap
| Concept | Summary |
|---|---|
| LCM | Smallest common multiple |
| HCF | Greatest common factor |
| Relation | HCF × LCM = Product of numbers |
| Co-prime Numbers | HCF = 1 |
| Useful For | Time, work, distribution, repetition problems |
10. Practice Questions
Test your knowledge with these practice questions. Click on "View Answer" to check your understanding.
Q1. LCM of 5 and 7 = ?
View Answer
35
Q2. HCF of 12 and 18 = ?
View Answer
6
Q3. If HCF = 5, LCM = 120, and one number = 15, find the other.
View Answer
(15 × x) = 5 × 120 → x = 40
Answer: 40
Q4. LCM of 8, 9, 12 = ?
View Answer
72
Q5. HCF of 42 and 63 = ?
View Answer
21
Q6. LCM of 15 and 20 = ?
View Answer
60
Q7. HCF of 27, 45, 63 = ?
View Answer
9
Q8. If two numbers are co-prime, then their HCF = ?
View Answer
1
Q9. LCM of two numbers is 72 and HCF is 6. If one number is 24, find the other.
View Answer
(24 × x) = 6 × 72 → x = 18
Answer: 18
Q10. HCF of 16 and 28 = ?
View Answer
4
✅ Exam Strategy Tip
🟢 Remember the key formula: HCF × LCM = Product of two numbers
🟢 For LCM: Take highest powers of all prime factors
🟢 For HCF: Take common prime factors with smallest powers
🟢 Practice prime factorization for quick calculations
You've completed LCM & HCF Concepts!
Courage Tip: Mastering LCM and HCF is essential for solving various quantitative aptitude problems. Remember that LCM helps find when events repeat together, while HCF helps in equal distribution scenarios. Practice the relationship formula HCF × LCM = Product of numbers as it frequently appears in competitive exams. Regular practice with different number combinations will build your speed and accuracy.
Master LCM & HCF for Competitive Exams!
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