GCD & LCM Calculator | Greatest Common Divisor & Least Common Multiple

📖 How to Use This GCD & LCM Calculator

Select GCD or LCM - Choose which calculation you need.

Enter numbers - Type or paste your numbers separated by commas or spaces.

Click "Calculate" - Get the result with a detailed step-by-step explanation.

🔢 What is GCD (Greatest Common Divisor)?

The Greatest Common Divisor (GCD), also known as the Greatest Common Factor (GCF) or Highest Common Factor (HCF), is the largest positive integer that divides each of the given numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 evenly.

📈 What is LCM (Least Common Multiple)?

The Least Common Multiple (LCM) is the smallest positive integer that is divisible by each of the given numbers. For example, the LCM of 4 and 6 is 12 because 12 is the smallest number that both 4 and 6 divide into evenly.

🧮 Methods to Find GCD and LCM

Euclidean Algorithm (GCD):
GCD(a,b) = GCD(b, a mod b) until b = 0.

Prime Factorization:
GCD = product of common prime factors (lowest power)
LCM = product of all prime factors (highest power)

Relation between GCD and LCM:
LCM(a,b) × GCD(a,b) = |a × b|

📊 Real-World Applications

Fraction simplification: GCD helps reduce fractions to lowest terms
Scheduling problems: LCM finds when events will occur simultaneously
Number theory: GCD is fundamental for modular arithmetic
Music theory: LCM helps find common time signatures
Gear ratios: GCD and LCM used in mechanical engineering

💡 Relationship Between GCD and LCM

For two numbers: LCM(a,b) × GCD(a,b) = a × b
Example: GCD(12,18)=6, LCM(12,18)=36 → 6×36=216=12×18
For more than two numbers, the relationship is more complex

❓ Frequently Asked Questions (FAQ)

What is the difference between GCD and HCF?

There is no difference! GCD (Greatest Common Divisor) and HCF (Highest Common Factor) are exactly the same concept - just different names.

How many numbers can I calculate GCD/LCM for?

Our calculator supports GCD and LCM for any number of positive integers. Just separate them with commas or spaces.

Does the Euclidean algorithm work for more than two numbers?

Yes! For multiple numbers, we calculate GCD(a,b) = g, then GCD(g,c) = GCD(a,b,c), and so on. Our calculator handles this automatically.

What if I enter zero?

Zero is divisible by any number. GCD(a,0) = |a|. LCM with zero is zero. Our calculator will handle zeros appropriately.

How do I find GCD using prime factorization?

Write each number as a product of primes. GCD is the product of common primes with the smallest exponent. For example, 12=2²×3, 18=2×3² → GCD=2×3=6.

What is the GCD of prime numbers?

The GCD of two different prime numbers is always 1 because they share no common factors. For example, GCD(7,11) = 1.