⭐ Sabse Important Points · एक नज़र में
- HCF Definition: The greatest number that is a common factor of all given numbers.
- LCM Definition: The least number which is exactly divisible by each of the given numbers.
- HCF by Prime Factorization: Express each number as a product of primes; HCF is the product of the smallest power of each common prime factor.
- LCM by Prime Factorization: Express each number as a product of primes; LCM is the product of the highest power of each prime factor present.
- HCF by Division Method: Divide the larger number by the smaller, then divide the divisor by the remainder, and continue until remainder is zero; the last divisor is the HCF.
- LCM of Fractions: LCM = (LCM of numerators) / (HCF of denominators).
Introduction · परिचय
LCM (Least Common Multiple) and HCF (Highest Common Factor) are fundamental concepts in number theory, frequently tested in RRB NTPC. They help solve problems related to divisibility, fractions, and time-speed-work. This short revision note covers definitions, methods, and key formulas with examples.
HCF: Highest Common Factor
HCF (also called GCD) is the largest number that divides two or more given numbers exactly without leaving a remainder. It is found by prime factorization or the division method.
📐 Formula / Rule Box · सूत्र याद रखो
HCF of Fractions
HCF(a/b, c/d) = HCF(a,c) / LCM(b,d)
⭐ Key Points · याद रखने वाली बातें
- HCF Definition: The greatest number that is a common factor of all given numbers.
- HCF by Prime Factorization: Express each number as a product of primes; HCF is the product of the smallest power of each common prime factor.
- HCF by Division Method: Divide the larger number by the smaller, then divide the divisor by the remainder, and continue until remainder is zero; the last divisor is the HCF.
- HCF of Fractions: HCF = (HCF of numerators) / (LCM of denominators).
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find the HCF of 36 and 48.
✅ Solution: 12
💡 Explanation: Prime factorization: 36 = 2^2 × 3^2, 48 = 2^4 × 3. Common prime factors: 2^2 and 3^1. HCF = 2^2 × 3 = 12.
LCM: Least Common Multiple
LCM is the smallest positive number that is a multiple of two or more given numbers. It is found by prime factorization or by the relationship with HCF.
📐 Formula / Rule Box · सूत्र याद रखो
LCM of Fractions
LCM(a/b, c/d) = LCM(a,c) / HCF(b,d)
HCF-LCM Relation
HCF(a,b) × LCM(a,b) = a × b
⭐ Key Points · याद रखने वाली बातें
- LCM Definition: The least number which is exactly divisible by each of the given numbers.
- LCM by Prime Factorization: Express each number as a product of primes; LCM is the product of the highest power of each prime factor present.
- LCM of Fractions: LCM = (LCM of numerators) / (HCF of denominators).
- Relation between HCF and LCM: For two numbers a and b, HCF(a,b) × LCM(a,b) = a × b.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find the LCM of 12 and 18.
✅ Solution: 36
💡 Explanation: Prime factorization: 12 = 2^2 × 3, 18 = 2 × 3^2. Highest powers: 2^2 and 3^2. LCM = 4 × 9 = 36.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1Find the HCF of 24 and 36.
👀 Answer dekho
✅ Answer: 12
💡 Explanation: 24 = 2^3 × 3, 36 = 2^2 × 3^2. Common factors: 2^2 and 3^1. HCF = 4 × 3 = 12.
Q2Find the LCM of 15 and 20.
👀 Answer dekho
✅ Answer: 60
💡 Explanation: 15 = 3 × 5, 20 = 2^2 × 5. Highest powers: 2^2, 3, 5. LCM = 4 × 3 × 5 = 60.
Q3The HCF of two numbers is 6 and their LCM is 36. If one number is 12, find the other.
👀 Answer dekho
✅ Answer: 18
💡 Explanation: Using HCF × LCM = product of numbers: 6 × 36 = 12 × x ⇒ 216 = 12x ⇒ x = 18.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Confusing HCF and LCM: HCF is the greatest common divisor, LCM is the least common multiple. Remember HCF ≤ numbers ≤ LCM.
- Forgetting to include the highest power of all prime factors when finding LCM; for HCF, use the lowest power of common primes only.
- In fraction problems, mixing up the formulas: HCF of fractions = HCF of numerators / LCM of denominators; LCM of fractions = LCM of numerators / HCF of denominators.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- HCF: Greatest number dividing all given numbers exactly; found by taking lowest powers of common prime factors.
- LCM: Smallest number divisible by all given numbers; found by taking highest powers of all prime factors.
- Key Relation: For two numbers, HCF × LCM = Product of the numbers.
- Fractions: HCF = HCF of numerators / LCM of denominators; LCM = LCM of numerators / HCF of denominators.
- Applications: HCF for dividing things into smaller equal parts; LCM for events repeating together.