Greatest Common Denominator Calculator
Find the largest number that divides all given numbers exactly
Calculate Greatest Common Denominator
Enter Numbers (up to 15)
Example Calculation
Find GCD of 48, 18, and 24
Method 1: Euclidean Algorithm
GCD(48, 18) = 6
GCD(6, 24) = 6
Result: GCD = 6
Method 2: Prime Factorization
48 = 2⁴ × 3¹
18 = 2¹ × 3²
24 = 2³ × 3¹
Common factors: 2¹ × 3¹ = 6
Result: GCD = 6
GCD Calculation Methods
Euclidean Algorithm
Uses division and remainders
Fast for large numbers
Prime Factorization
Breaks numbers into prime factors
Shows mathematical structure
List Factors
Find all common factors
Intuitive for small numbers
GCD Quick Tips
GCD is always positive, even for negative numbers
GCD(a, 0) = |a| for any non-zero number a
GCD of coprime numbers is always 1
Used to simplify fractions to lowest terms
Understanding Greatest Common Denominator (GCD)
What is the GCD?
The Greatest Common Denominator (also called Greatest Common Divisor) is the largest positive integer that divides all numbers in a set without leaving a remainder. It's fundamental in number theory and has practical applications in simplifying fractions, finding patterns, and solving mathematical problems.
Why is it Important?
- •Simplifies fractions to their lowest terms
- •Solves problems involving ratios and proportions
- •Used in cryptography and computer algorithms
- •Helps in finding patterns and tessellations
Euclidean Algorithm
GCD(a, b) = GCD(b, a mod b)
Continue until remainder = 0
Prime Factorization Method
- 1. Find prime factorization of each number
- 2. Identify common prime factors
- 3. Take lowest power of each common prime
- 4. Multiply these together to get GCD
Example: GCD(12, 18) = 6
12 = 2² × 3¹, 18 = 2¹ × 3²
Common: 2¹ × 3¹ = 6