Greatest Common Divisor Calculator
Find the largest positive integer that divides all given numbers without remainder
Calculate Greatest Common Divisor
Enter Numbers (up to 15)
Example Calculation
Find GCD of 15, 45, and 75
Method 1: List of Factors
Factors of 15: 1, 3, 5, 15
Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 75: 1, 3, 5, 15, 25, 75
Common factors: 1, 3, 5, 15
Result: GCD = 15
Method 2: Prime Factorization
15 = 3¹ × 5¹
45 = 3² × 5¹
75 = 3¹ × 5²
Common factors: 3¹ × 5¹ = 15
Result: GCD = 15
GCD Applications
Fraction Simplification
Reduce fractions to lowest terms
Divide numerator and denominator by GCD
Cryptography
RSA algorithm security
Public key encryption systems
Engineering
Gear ratios and mechanical systems
Precise part relationships
GCD Quick Facts
GCD is always positive, regardless of input signs
GCD(a, b) × LCM(a, b) = a × b for two numbers
Numbers with GCD = 1 are called coprime
Euclidean algorithm is most efficient for large numbers
Understanding Greatest Common Divisor (GCD)
What is the GCD?
The Greatest Common Divisor (GCD), also known as the Greatest Common Factor (GCF), is the largest positive integer that divides all numbers in a set without leaving a remainder. It's a fundamental concept in number theory with wide applications in mathematics, computer science, and engineering.
Key Properties
- •Always positive, even for negative inputs
- •GCD(a, b) ≤ min(a, b) for positive integers
- •If GCD(a, b) = 1, then a and b are coprime
- •GCD is commutative: GCD(a, b) = GCD(b, a)
Calculation Methods
1. Euclidean Algorithm
Most efficient method using repeated division:
GCD(a, b) = GCD(b, a mod b)
2. Prime Factorization
Find prime factors and take the lowest power of each common prime factor.
3. List of Factors
List all factors of each number and find the largest common one.