Mathematics: Understanding HCF and LCM
Understanding HCF and LCM is fundamental for solving problems in arithmetic, fractions, and algebra. Before diving into the methods, let's review some basic number types.
🔍 Prerequisites (बुनियादी अवधारणाएं)
1. Prime Numbers (अभाज्य संख्या)
Numbers that have exactly two factors: 1 and the number itself.
- Example: 2, 3, 5, 7, 11, 13...
- Note: 2 is the only even prime number.
2. Composite Numbers (भाज्य संख्या)
Numbers that have more than two factors.
- Example: 4, 6, 8, 9, 10, 12...
3. Co-Prime Numbers (सह-अभाज्य संख्या)
Two numbers that have only 1 as a common factor.
- Example: (3, 4), (5, 9), (10, 11).
🔹 HCF (Highest Common Factor)
HCF is the largest number that divides two or more numbers exactly. It is also known as GCD (Greatest Common Divisor).
Method 1: Prime Factorization
- Find the prime factors of each number.
- Identify the common factors.
- Multiply the common factors to get the HCF.
Example: HCF of 6 and 12
- 6 = 2 × 3
- 12 = 2 × 2 × 3
- Common factors: 2 and 3
- HCF = 2 × 3 = 6
Method 2: Long Division (Continuous Division)
Used for large numbers where prime factorization is difficult.
- Divide the larger number by the smaller one.
- Use the remainder as the new divisor and the previous divisor as the new dividend.
- Repeat until the remainder is 0. The last divisor is the HCF.
Example: HCF of 24 and 36
- 36 ÷ 24 = 1 (Remainder 12)
- 24 ÷ 12 = 2 (Remainder 0)
- HCF = 12
🔸 LCM (Least Common Multiple)
LCM is the smallest number that is a multiple of two or more numbers.
Method 1: Prime Factorization
- Find prime factors of each number.
- Multiply the highest power of every prime factor present.
Example: LCM of 6 and 12
- 6 = 2 × 3
- 12 = 2² × 3
- LCM = 2² × 3 = 12
Method 2: Common Division Method
Divide all numbers by prime factors simultaneously until all quotients are 1.
💡 The Important Relationship
For any two numbers 'a' and 'b':
Product of two numbers = HCF × LCM(a × b) = HCF(a, b) × LCM(a, b)
Verification:
- Numbers: 6 and 12
- Product: 6 × 12 = 72
- HCF (6) × LCM (12) = 72