GCD Calculator

💡 Expert Tips

Euclidean Algorithm is Ancient & Fast

Euclid described this algorithm 2,300 years ago! It's incredibly efficient—even for huge numbers. Modern computers use it for cryptography.

GCD × LCM = Product

Here's a cool relationship: GCD(a,b) × LCM(a,b) = a × b. So if you know GCD, you can easily find LCM.

Simplifying Fractions

To simplify a fraction like 12/18, divide both the numerator and denominator by their GCD (6). Result: 2/3. That's the lowest terms!

⚠️ Common Mistakes

Confusing GCD with LCM

GCD is the LARGEST divisor. LCM is the SMALLEST multiple. Opposite concepts! GCD(12, 18) = 6. LCM(12, 18) = 36.

Listing All Factors Manually

For large numbers, listing all factors is tedious and error-prone. Use the Euclidean algorithm instead—it's much faster.

Forgetting GCD(a,0) = a

Any number's GCD with 0 is the number itself. This is a special case that trips people up.

❓ Frequently Asked Questions

What is GCD? +

GCD (Greatest Common Divisor) is the largest positive integer that divides both numbers without a remainder. It's also called HCF (Highest Common Factor).

How do you calculate GCD? +

The most efficient method is the Euclidean algorithm: repeatedly divide the larger number by the smaller and replace the larger with the remainder until the remainder is 0.

What is the GCD of two prime numbers? +

The GCD of two different prime numbers is always 1, because prime numbers have no common divisors except 1.

What is the difference between GCD and LCM? +

GCD is the largest number that divides both numbers, while LCM (Least Common Multiple) is the smallest number that both numbers divide into. They are related: GCD(a,b) × LCM(a,b) = a × b.

When is GCD used in real life? +

GCD is used to simplify fractions, find common denominators, solve Diophantine equations, and in cryptography (RSA algorithm).

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