GCD Calculator
Calculate the Greatest Common Divisor with step-by-step explanations using multiple algorithms
Calculate GCD
GCD Result
Common GCD Examples
Algorithms
Euclidean Algorithm
Fast, uses division and remainder operations
Prime Factorization
Shows the mathematical structure, good for learning
GCD Properties
GCD is always positive for positive integers
If GCD = 1, numbers are coprime (relatively prime)
GCD(a,b) × LCM(a,b) = a × b
GCD is commutative: GCD(a,b) = GCD(b,a)
Used to simplify fractions to lowest terms
Understanding Greatest Common Divisor (GCD)
What is GCD?
The Greatest Common Divisor (GCD) of a set of numbers is the largest positive integer that divides each number in the set without leaving a remainder.
Key Properties
- Positive: GCD is always positive for positive integers
- Divisibility: GCD divides all numbers in the set
- Maximum: No larger number can divide all inputs
- Unique: There is exactly one GCD for any set
Calculation Methods
1. Euclidean Algorithm
Uses repeated division: GCD(a,b) = GCD(b, a mod b)
Example: GCD(48, 18)
48 = 18 × 2 + 12
18 = 12 × 1 + 6
12 = 6 × 2 + 0
GCD = 6
2. Prime Factorization
Find common prime factors and use lowest powers
Example: GCD(48, 18)
48 = 2⁴ × 3¹
18 = 2¹ × 3²
GCD = 2¹ × 3¹ = 6
Real-World Applications
Fraction Simplification: Use GCD to reduce fractions to lowest terms
Tile Problems: Find the largest square tile that can cover a rectangular area
Gear Systems: Determine gear ratios and mechanical systems
Music Theory: Calculate harmonic intervals and frequency relationships