Chapter 4: Operations Based on HCF and LCM
Subject: Mathematics | Language: English | For: Competitive Exam Aspirants
HCF (Highest Common Factor) and LCM (Lowest Common Multiple) are fundamental concepts in mathematics that are essential for competitive exams like IBPS, SSC, Railway, and other government exams. These concepts help in solving various mathematical problems efficiently and are frequently tested in aptitude sections.
HCF (Highest Common Factor)
Definition: The HCF of two or more numbers is the greatest number that divides each of them exactly, without leaving any remainder.
| Method | Steps | Example |
|---|---|---|
| Prime Factorization | Express numbers as product of primes, take common factors with lowest power | 12 = 2²×3, 18 = 2×3² HCF = 2×3 = 6 |
| Division Method | Divide larger by smaller, continue with remainder | 48 ÷ 18 = 2 remainder 12 18 ÷ 12 = 1 remainder 6 12 ÷ 6 = 2 remainder 0 HCF = 6 |
| Listing Factors | List all factors, find highest common | Factors of 12: 1,2,3,4,6,12 Factors of 18: 1,2,3,6,9,18 Common: 1,2,3,6 HCF = 6 |
LCM (Lowest Common Multiple)
Definition: The LCM of two or more numbers is the smallest number that is a multiple of each of them.
| Method | Steps | Example |
|---|---|---|
| Prime Factorization | Express numbers as product of primes, take all factors with highest power | 12 = 2²×3, 18 = 2×3² LCM = 2²×3² = 36 |
| Division Method | Divide by common factors, multiply all divisors and remainders | 2|12,18 3|6,9 2,3 LCM = 2×3×2×3 = 36 |
| Listing Multiples | List multiples, find smallest common | Multiples of 12: 12,24,36,48... Multiples of 18: 18,36,54... Common: 36,72... LCM = 36 |
Important Properties
HCF Properties
- HCF of co-prime numbers is 1
- HCF ≤ each of the numbers
- HCF of two numbers divides their LCM
- If HCF(a,b) = 1, then a and b are co-prime
LCM Properties
- LCM ≥ each of the numbers
- LCM of co-prime numbers = their product
- LCM is always divisible by HCF
- For two numbers: HCF × LCM = Product
Important Formula: For any two numbers a and b, HCF(a,b) × LCM(a,b) = a × b
Solved Examples
Solution:
Prime factorization method:
24 = 2³ × 3¹
36 = 2² × 3²
HCF = 2² × 3¹ = 4 × 3 = 12
LCM = 2³ × 3² = 8 × 9 = 72
Verification: HCF × LCM = 12 × 72 = 864 = 24 × 36 ✓
Solution:
Let the numbers be a and b.
HCF(a,b) = 8
LCM(a,b) = 96
Product = a × b = 8 × 96 = 768
Possible pairs: (8,96), (16,48), (24,32), (32,24), (48,16), (96,8)
Answer: The numbers are 24 and 32 (or 32 and 24)
Solution:
Required number = LCM(12,16,18) + 5
12 = 2² × 3
16 = 2⁴
18 = 2 × 3²
LCM = 2⁴ × 3² = 16 × 9 = 144
Answer: 144 + 5 = 149
Quick Tip: For three numbers a, b, c: LCM(a,b,c) = LCM(LCM(a,b), c) and HCF(a,b,c) = HCF(HCF(a,b), c)
Practice Questions
Exercise – 1: Solve these questions to test your understanding
Q1. What is the HCF of 48 and 72?
(a) 12 (b) 16 (c) 24 (d) 36
Q2. What is the LCM of 15 and 25?
(a) 50 (b) 75 (c) 100 (d) 125
Q3. If HCF of two numbers is 6 and their product is 180, what is their LCM?
(a) 15 (b) 20 (c) 30 (d) 45
Q4. Which of the following pairs are co-prime?
(a) 8, 15 (b) 12, 18 (c) 14, 21 (d) 9, 27
Q5. The LCM of 6, 8, and 12 is:
(a) 12 (b) 24 (c) 48 (d) 72
Q6. The HCF of 36, 60, and 84 is:
(a) 6 (b) 12 (c) 18 (d) 24
Advanced Questions
Q7. If LCM of two numbers is 120 and their HCF is 10, and one number is 20, what is the other?
(a) 30 (b) 40 (c) 50 (d) 60
Q8. The HCF of two consecutive numbers is always:
(a) 1 (b) 2 (c) 3 (d) 4
Q9. Find the least number which when divided by 8, 12, and 16 leaves remainder 3:
(a) 51 (b) 99 (c) 147 (d) 195
Finding HCF of Large numbers using Division method (Euclidean).
Q1. What is the HCF of 9876 and 5432?
(a) 2 (b) 4 (c) 6 (d) 8
Q2. What is the HCF of 12345 and 6789?
(a) 1 (b) 2 (c) 3 (d) 5
Q3. What is the HCF of 15678 and 23456?
(a) 2 (b) 4 (c) 6 (d) 8
Q4. What is the HCF of 19284 and 14256?
(a) 12 (b) 24 (c) 36 (d) 48
Q5. What is the HCF of 28476 and 19638?
(a) 6 (b) 18 (c) 42 (d) 54
Q6. What is the HCF of 35000 and 28750?
(a) 250 (b) 500 (c) 750 (d) 1000
Q7. What is the HCF of 46200 and 38500?
(a) 100 (b) 200 (c) 300 (d) 400
Q8. What is the HCF of 50412 and 37890?
(a) 6 (b) 12 (c) 18 (d) 24
Q9. What is the HCF of 72930 and 54675?
(a) 15 (b) 45 (c) 105 (d) 315
Q10. What is the HCF of 81234 and 56789?
(a) 1 (b) 3 (c) 7 (d) 13
Quick Tips for Competitive Exams
For HCF:
- Use prime factorization for accuracy
- Division method is faster for large numbers
- HCF of co-primes is always 1
- HCF ≤ smallest number
For LCM:
- LCM ≥ largest number
- For co-primes, LCM = product
- Use the formula: HCF × LCM = Product
- For three numbers: LCM(a,b,c) = LCM(LCM(a,b), c)
Conclusion: Mastery of HCF and LCM is crucial for competitive exams. Practice regularly with different types of questions to improve speed and accuracy. Remember the key formula: HCF × LCM = Product of two numbers.
Related Topics
📚 Keep practicing! Mastery of HCF and LCM is essential for competitive exams like JNV & Sainik School Entrance, IBPS, SSC, and Railways.
