⭐ Sabse Important Points · एक नज़र में
- HCF Definition: The largest positive integer that divides two or more numbers without leaving a remainder.
- Prime Factorization Method: Express each number as a product of primes; HCF = product of smallest powers of common primes; LCM = product of greatest powers of all primes.
- LCM Definition: The smallest positive integer that is divisible by two or more numbers.
- Division Method for HCF: Divide larger by smaller, then divide divisor by remainder, repeat until remainder is 0; last divisor is HCF.
- Product Rule: For two numbers a and b, HCF(a,b) × LCM(a,b) = a × b. This holds only for two numbers.
- LCM via HCF: For two numbers, LCM = (a × b) / HCF(a,b).
Introduction · परिचय
LCM (Least Common Multiple) and HCF (Highest Common Factor) are fundamental concepts in number theory, frequently tested in RRB Group D Mathematics. They help solve problems related to divisibility, fractions, and time-work scenarios. This short revision note covers definitions, methods, key formulas, and common exam patterns.
HCF (Highest Common Factor) and LCM (Least Common Multiple)
HCF of two or more numbers is the greatest number that divides each of them exactly. LCM is the smallest number that is exactly divisible by each of them. For two numbers a and b, HCF × LCM = a × b. HCF is also called GCD (Greatest Common Divisor).
📐 Formula / Rule Box · सूत्र याद रखो
Product Rule (two numbers)
HCF(a,b) × LCM(a,b) = a × b
HCF of fractions
HCF = HCF of numerators / LCM of denominators
LCM of fractions
LCM = LCM of numerators / HCF of denominators
Product Rule
For two numbers a and b, HCF(a,b) × LCM(a,b) = a × b. This holds only for two numbers.
| HCF | LCM |
|---|---|
| Greatest common divisor | Least common multiple |
| Always ≤ smallest number | Always ≥ largest number |
| Used to split into equal groups | Used to find common meeting points |
| Found by taking lowest powers of common prime factors | Found by taking highest powers of all prime factors |
⭐ Key Points · याद रखने वाली बातें
- HCF Definition: The largest positive integer that divides two or more numbers without leaving a remainder.
- LCM Definition: The smallest positive integer that is divisible by two or more numbers.
- Co-prime Numbers: If HCF of two numbers is 1, they are co-prime; their LCM is their product.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find HCF and LCM of 12 and 18.
✅ Solution: HCF = 6, LCM = 36
💡 Explanation: Prime factors: 12 = 2² × 3, 18 = 2 × 3². HCF = product of lowest powers of common factors = 2¹ × 3¹ = 6. LCM = product of highest powers of all factors = 2² × 3² = 36. Check: 6 × 36 = 216 = 12 × 18.
Methods to Find HCF and LCM
Two main methods are used: Prime Factorization and Division Method. For HCF, prime factorization takes the product of lowest powers of common prime factors; for LCM, take the product of highest powers of all prime factors. The division method (Euclid's algorithm) is efficient for large numbers.
- Take two numbers
- Divide larger by smaller
- Note remainder
- If remainder = 0, divisor is HCF
- Else divide divisor by remainder
- Repeat until remainder = 0
⭐ Key Points · याद रखने वाली बातें
- Prime Factorization Method: Express each number as a product of primes; HCF = product of smallest powers of common primes; LCM = product of greatest powers of all primes.
- Division Method for HCF: Divide larger by smaller, then divide divisor by remainder, repeat until remainder is 0; last divisor is HCF.
- LCM via HCF: For two numbers, LCM = (a × b) / HCF(a,b).
- Multiple Numbers: HCF of more than two numbers is found by repeated HCF; LCM by repeated LCM.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find HCF of 48 and 60 using division method.
✅ Solution: HCF = 12
💡 Explanation: 60 ÷ 48 = 1 remainder 12. 48 ÷ 12 = 4 remainder 0. Last divisor = 12, so HCF = 12.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: Find LCM of 15, 20, and 30.
✅ Solution: LCM = 60
💡 Explanation: Prime factors: 15 = 3 × 5, 20 = 2² × 5, 30 = 2 × 3 × 5. LCM = 2² × 3 × 5 = 60.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1Find the HCF of 36 and 84.
👀 Answer dekho
✅ Answer: 12
💡 Explanation: 36 = 2² × 3², 84 = 2² × 3 × 7. Common factors with lowest powers: 2² × 3 = 12.
Q2The LCM of two numbers is 120 and their HCF is 4. If one number is 24, find the other.
👀 Answer dekho
✅ Answer: 20
💡 Explanation: Using HCF × LCM = product: 4 × 120 = 24 × x ⇒ 480 = 24x ⇒ x = 20.
Q3Find the LCM of fractions 2/3 and 4/9.
👀 Answer dekho
✅ Answer: 4/3
💡 Explanation: LCM of fractions = LCM of numerators / HCF of denominators = LCM(2,4) / HCF(3,9) = 4 / 3.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Confusing HCF and LCM: Remember HCF divides, LCM is divisible.
- Using product rule for more than two numbers: HCF × LCM = product holds only for two numbers.
- Forgetting to reduce fractions before finding HCF/LCM: Always simplify fractions first.
- In division method, stopping before remainder becomes zero: Continue until remainder is 0; last divisor is HCF.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- HCF: Greatest number dividing all given numbers; LCM: Smallest number divisible by all given numbers.
- Product Rule: For two numbers, HCF × LCM = product of numbers.
- Fractions: HCF = HCF of numerators / LCM of denominators; LCM = LCM of numerators / HCF of denominators.
- Co-prime numbers: HCF = 1, LCM = product of numbers.
- Division method: Repeated division until remainder 0 gives HCF.