kgV und ggT Rechner
Kleinstes gemeinsames Vielfaches und groessten gemeinsamen Teiler sofort berechnen
LCM & GCD Calculator
Find the Least Common Multiple and Greatest Common Divisor
Enter two positive integers
LCM(a,b) = (a x b) / GCD(a,b)What is an LCM & GCD Calculator?
An LCM & GCD Calculator is a math tool that finds two important number relationships:
- GCD (Greatest Common Divisor) -- the largest positive integer that divides each number with no remainder
- LCM (Least Common Multiple) -- the smallest positive integer that is a multiple of each number
These concepts are used constantly in arithmetic, fractions, and algebra. GCD is most commonly used to simplify fractions and reduce ratios. LCM is most commonly used to find common denominators, combine fractions, and solve problems involving repeating schedules or cycles.
Because finding LCM and GCD by hand can be slow—especially for large numbers or multiple values—this calculator lets you enter your numbers and get the correct result instantly.
How to Use This LCM & GCD Calculator
- Enter your numbers -- input two positive integers into the fields above
- Click 'Calculate' -- to compute both GCD and LCM
- Review both results -- the calculator displays the GCD and LCM simultaneously
- Use the output -- apply the GCD for fraction simplification or the LCM for common denominators, scheduling, or math problems
Tips:
- Use whole positive numbers for the most standard interpretation of LCM and GCD
- If you include 0, the behavior may differ depending on the definition used (many tools define gcd(a, 0) = |a| and lcm(a, 0) = 0)
- For multiple numbers, calculators typically compute results by combining them step-by-step (pairwise)
Formulas
Greatest Common Divisor (GCD)
The GCD of two numbers a and b is the largest integer that divides both without remainder. A common method is the Euclidean Algorithm:
- Divide: a = bq + r
- Replace: a ← b, b ← r
- Repeat until r = 0
The last non-zero remainder is the GCD. In short:
gcd(a, b) = gcd(b, a mod b)
until the remainder is 0
Least Common Multiple (LCM)
For two non-zero integers:
lcm(a, b) = |a × b| / gcd(a, b)
Use the GCD to compute LCM efficiently
More than Two Numbers
For multiple values, compute pairwise:
gcd(a, b, c) = gcd(gcd(a, b), c)
lcm(a, b, c) = lcm(lcm(a, b), c)
Example Calculations
Example 1: Find GCD of 48 and 18
48 factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
18 factors: 1, 2, 3, 6, 9, 18
Largest common factor: 6
Result: gcd(48, 18) = 6
Example 2: Find LCM of 12 and 18
First find GCD: gcd(12, 18) = 6
Calculation: lcm(12, 18) = |12 × 18| / 6 = 216 / 6 = 36
Result: lcm(12, 18) = 36
Example 3: Simplify a Fraction Using GCD
Problem: Simplify 84/126
GCD: gcd(84, 126) = 42
Calculation: 84 ÷ 42 = 2, 126 ÷ 42 = 3
Result: 84/126 simplifies to 2/3
Example 4: LCM for Multiple Numbers
Problem: Find LCM of 4, 6, 10
Step 1: lcm(4, 6) = 12
Step 2: lcm(12, 10) = 60
Result: lcm(4, 6, 10) = 60
Frequently Asked Questions
What’s the difference between GCD and LCM?
GCD is the largest number that divides both numbers. LCM is the smallest number that both numbers divide into (the smallest shared multiple). GCD helps simplify; LCM helps combine or align values.
When do I use GCD?
Use GCD when simplifying fractions, reducing ratios, or finding the largest equal group size (for example, splitting items evenly).
When do I use LCM?
Use LCM when finding a common denominator for fractions, aligning repeating schedules (like events every 6 and 8 days), or solving problems involving cycles.
Can LCM & GCD be calculated for more than two numbers?
Yes. The standard approach is to compute the result pairwise (combine two numbers at a time) until all numbers are included.
What happens if one of the numbers is 0?
Many definitions treat gcd(a, 0) = |a| and lcm(a, 0) = 0, but calculators may vary. If your tool allows 0, it should clearly define how it handles it.
Want to add this lcm & gcd calculator to your website? Get a custom embed code that matches your site's design and keeps visitors engaged.
Was sind kgV und ggT?
Der groesste gemeinsame Teiler (ggT) ist die groesste Zahl, die zwei oder mehr ganze Zahlen ohne Rest teilt. Das kleinste gemeinsame Vielfache (kgV) ist die kleinste positive Zahl, die ein Vielfaches aller angegebenen Zahlen ist. Beide Begriffe sind grundlegend in der Zahlentheorie und der Bruchrechnung und begegnen dir immer wieder in der Mathematik.
Der ggT wird verwendet, um Brueche zu kuerzen: Teile Zaehler und Nenner durch den ggT, um den Bruch auf seinen kleinsten Ausdruck zu bringen. Das kgV wird benoetigt, um beim Addieren oder Subtrahieren von Bruechen mit verschiedenen Nennern einen gemeinsamen Nenner zu finden. Dieser Rechner findet beide Werte sofort mit dem euklidischen Algorithmus.
So verwendest du den kgV- und ggT-Rechner
- Gib zwei oder mehr ganze Zahlen in die Eingabefelder ein.
- Klicke auf Berechnen.
- Lese die angezeigten Ergebnisse fuer ggT und kgV ab.
- Verwende den ggT zum Kuerzen von Bruechen oder das kgV zum Finden eines gemeinsamen Nenners.
Formeln und Algorithmen
Euklidischer Algorithmus (ggT):
ggT(a, b) = ggT(b, a mod b) bis b = 0
Beispiel: ggT(48, 18) -> ggT(18, 12) -> ggT(12, 6) -> ggT(6, 0) = 6
kgV aus dem ggT:
kgV(a, b) = |a x b| / ggT(a, b)
Beispiel: kgV(4, 6) = 24 / 2 = 12
Primfaktorzerlegung:
48 = 2^4 x 3, 18 = 2 x 3^2
ggT = 2^1 x 3^1 = 6, kgV = 2^4 x 3^2 = 144kgV(a, b) x ggT(a, b) = a x b. Diese Identitaet bietet eine schnelle Abkuerzung, sobald du einen der beiden Werte kennst.
Loesungsbeispiele
Beispiel 1: ggT(12, 8) und kgV(12, 8)
ggT(12, 8): 12 mod 8 = 4, dann ggT(8, 4) = 4. Also ggT = 4. kgV = (12 x 8) / 4 = 96 / 4 = 24.
Beispiel 2: Bruch 18/24 kuerzen
ggT(18, 24) berechnen: 24 mod 18 = 6, dann ggT(18, 6) = 6. Beide durch 6 teilen: 18/24 = 3/4.
Beispiel 3: 1/4 + 1/6 addieren
kgV(4, 6) = 12 berechnen. Umschreiben: 1/4 = 3/12 und 1/6 = 2/12. Addieren: 3/12 + 2/12 = 5/12.