Example 1
\(\text{GCD}(12, 18) = 6, \text{LCM}(12, 18) = 36\)
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GCD and LCM
The Greatest Common Divisor (GCD) and Least Common Multiple (LCM) are fundamental concepts in number theory with practical applications in fractions, ratios, and problem-solving.
GCD (Greatest Common Divisor):
- The largest positive integer that divides each of the numbers
- Also called HCF (Highest Common Factor)
- Used for simplifying fractions
- Euclidean algorithm: GCD(a,b) = GCD(b, a mod b)
LCM (Least Common Multiple):
- The smallest positive integer divisible by all numbers
- Used for adding/subtracting fractions with different denominators
- Formula: LCM(a,b) = (a × b) / GCD(a,b)
- Can be found using prime factorization
Relationship:
- GCD(a,b) × LCM(a,b) = a × b (for two numbers)
- GCD divides both numbers, both numbers divide LCM
- If GCD(a,b) = 1, the numbers are coprime
Worked Examples
Example 2
\(\text{GCD}(24, 36, 48) = 12, \text{LCM}(24, 36, 48) = 144\)