Example 1
\(\text{GCD}(12, 18) = 6, \text{LCM}(12, 18) = 36\)
KGV und GGT-Rechner
Berechnen Sie das kleinste gemeinsame Vielfache (KGV) und den größten gemeinsamen Teiler (GGT) von zwei oder mehr Zahlen mit detaillierten Schritt-für-Schritt-Lösungen
Ergebnisse
Geben Sie Werte ein und klicken Sie auf Berechnen, um das Ergebnis zu sehen.
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
\(\text{GCD}(24, 36, 48) = 12, \text{LCM}(24, 36, 48) = 144\)