What is the Greatest Common Divisor (GCD)?

The GCD is the largest positive integer that divides both numbers without a remainder. For example, the GCD of 48 and 18 is 6.

What does "relatively prime" mean?

Two numbers are relatively prime (or coprime) if their GCD is 1, meaning they share no common factors other than 1.

How does the Euclidean algorithm work?

The Euclidean algorithm repeatedly applies the division algorithm: divide the larger number by the smaller, then replace the larger with the smaller and the smaller with the remainder, until the remainder is 0.

GCD is used in fraction simplification, finding lowest common denominators, cryptography, and many areas of mathematics and computer science.

Math Calculators

Greatest Common Divisor Calculator

Find the GCD (Greatest Common Divisor) of two or more numbers using efficient mathematical algorithms including Euclidean method for number theory, algebra, and mathematical problem solving with step-by-step explanations and multiple calculation methods.

How to Use the Greatest Common Divisor Calculator

Use the Greatest Common Divisor Calculator to the GCD (Greatest Common Divisor) of two or more numbers using efficient mathematical algorithms including Euclidean method for number theory, algebra, and mathematical problem solving with step-by-step explanations and multiple calculation methods.. Enter your values to get accurate, instant results tailored to your situation.

Free math calculators for algebra, geometry, statistics, and more. Solve complex mathematical problems with step-by-step solutions.

Common Uses

Calculate greatest accurately and instantly
Calculate common accurately and instantly
Calculate divisor accurately and instantly
Calculate find accurately and instantly
Calculate gcd accurately and instantly

Related Calculators

More Math Calculators

Browse all 311+ free online calculators

GCD Guide

Finding common divisors

Expert Tips

GCD = Largest Number Dividing Both: GCD(12, 18) = 6 (divides both evenly). GCD(15, 25) = 5. GCD(7, 11) = 1 (coprime, no common factors). Used to simplify fractions: 12/18 ÷ GCD(6) = 2/3.
Euclidean Algorithm: Faster for Large Numbers: GCD(48, 18): 48 = 18×2 + 12, then GCD(18, 12): 18 = 12×1 + 6, then GCD(12, 6): 12 = 6×2 + 0. Answer: 6. Recursive: GCD(a,b) = GCD(b, a mod b) until remainder = 0.
Prime Factorization Method: 24 = 2³ × 3, 36 = 2² × 3². GCD = product of lowest powers of common primes = 2² × 3 = 12. Works but slower than Euclidean for large numbers. Good for understanding.
GCD(a, b) × LCM(a, b) = a × b: Useful identity. If GCD(12, 18) = 6, then LCM(12, 18) = 12×18 / 6 = 36. Can find one from the other. Shortcut for related calculations.

Essential Fundamentals — GCD methods

Calculation Methods

Listing Factors Method: List all factors of each number. 12: {1,2,3,4,6,12}. 18: {1,2,3,6,9,18}. Common: {1,2,3,6}. Greatest: 6. Simple but tedious for large numbers.
Euclidean Algorithm: Most efficient. Divide larger by smaller, repeat with remainder. GCD(60, 24): 60÷24 = 2 R 12. GCD(24, 12): 24÷12 = 2 R 0. GCD = 12. Terminates quickly even for huge numbers.

Advanced Strategies — Optimization methods

Efficiency Techniques

Multiple Numbers: Find GCD of multiple numbers sequentially: GCD(12,18,24) = GCD(GCD(12,18),24) = GCD(6,24) = 6. Or use prime factorization: take lowest power of each common prime.
Property Applications: GCD(a,b) = GCD(a-b,b) for simplification. GCD(100,64) = GCD(36,64) = GCD(36,28) continues until manageable. Useful when Euclidean division difficult mentally.

Frequently Asked Questions

What is the Greatest Common Divisor (GCD)?: The GCD is the largest positive integer that divides both numbers without a remainder. For example, the GCD of 48 and 18 is 6.
What does "relatively prime" mean?: Two numbers are relatively prime (or coprime) if their GCD is 1, meaning they share no common factors other than 1.
How does the Euclidean algorithm work?: The Euclidean algorithm repeatedly applies the division algorithm: divide the larger number by the smaller, then replace the larger with the smaller and the smaller with the remainder, until the remainder is 0.
Why is GCD useful?: GCD is used in fraction simplification, finding lowest common denominators, cryptography, and many areas of mathematics and computer science.