Calculadora de MMC e MDC
Encontre o Mínimo Múltiplo Comum e o Máximo Divisor Comum na hora
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.
O que são MMC e MDC?
O Máximo Divisor Comum (MDC) é o maior número que divide dois ou mais números sem deixar resto. O Mínimo Múltiplo Comum (MMC) é o menor número positivo que é múltiplo de todos os números dados. Ambos são conceitos fundamentais em teoria dos números e na aritmética de frações, e você vai usá-los o tempo todo ao longo da sua jornada nas matemáticas.
O MDC é usado para simplificar frações: divida o numerador e o denominador pelo MDC para obter a fração na forma irredutível. O MMC é usado para encontrar o denominador comum ao somar ou subtrair frações com denominadores diferentes. Esta calculadora encontra os dois valores instantaneamente usando o algoritmo de Euclides.
Como usar a calculadora de MMC e MDC
- Insira dois ou mais números inteiros nos campos de entrada.
- Clique em Calcular.
- Leia os resultados do MDC e do MMC exibidos abaixo.
- Use o MDC para simplificar frações ou o MMC para encontrar um denominador comum.
Fórmulas e algoritmos
Algoritmo de Euclides (MDC):
MDC(a, b) = MDC(b, a mod b) até que b = 0
Exemplo: MDC(48, 18) → MDC(18, 12) → MDC(12, 6) → MDC(6, 0) = 6
MMC a partir do MDC:
MMC(a, b) = |a × b| / MDC(a, b)
Exemplo: MMC(4, 6) = 24 / 2 = 12
Método da fatoração prima:
48 = 2⁴ × 3, 18 = 2 × 3²
MDC = 2¹ × 3¹ = 6, MMC = 2⁴ × 3² = 144MMC(a, b) × MDC(a, b) = a × b. Essa identidade oferece um atalho rápido assim que você conhece um dos dois valores.
Exemplos resolvidos
Exemplo 1: MDC(12, 8) e MMC(12, 8)
MDC(12, 8): 12 mod 8 = 4, depois MDC(8, 4) = 4. Portanto MDC = 4. MMC = (12 × 8) / 4 = 96 / 4 = 24.
Exemplo 2: Simplificar a fração 18/24
Encontrar MDC(18, 24): 24 mod 18 = 6, depois MDC(18, 6) = 6. Dividir os dois por 6: 18/24 = 3/4.
Exemplo 3: Somar 1/4 + 1/6
Encontrar MMC(4, 6) = 12. Reescrever: 1/4 = 3/12 e 1/6 = 2/12. Somar: 3/12 + 2/12 = 5/12.