Euclidean Algorithm Calculator
Find the Greatest Common Divisor (GCD) using the Euclidean Algorithm with step-by-step solutions
Calculate GCD using Euclidean Algorithm
Enter a positive integer
Enter a positive integer
Uses remainder division for efficient calculation
Example Problems
Example 1: GCD(48, 18)
Modulo Method:
GCD(48, 18) = GCD(18, 48 mod 18) = GCD(18, 12)
GCD(18, 12) = GCD(12, 18 mod 12) = GCD(12, 6)
GCD(12, 6) = GCD(6, 12 mod 6) = GCD(6, 0) = 6
Answer: GCD(48, 18) = 6
Example 2: GCD(462, 364)
Modulo Method (Efficient):
GCD(462, 364) = GCD(364, 98) = GCD(98, 70) = GCD(70, 28) = GCD(28, 14) = GCD(14, 0) = 14
Answer: GCD(462, 364) = 14
Subtraction method would take many more steps for these numbers
Example 3: GCD(17, 13)
Two Prime Numbers:
GCD(17, 13) = GCD(13, 4) = GCD(4, 1) = GCD(1, 0) = 1
Answer: GCD(17, 13) = 1 (coprime numbers)
Algorithm Methods
Modulo Method
Uses remainder division - fast and efficient
Subtraction Method
Uses repeated subtraction - easier to understand
Properties
Both methods are mathematically equivalent
Key Concepts
GCD Properties
GCD(a,b) = GCD(b, a mod b)
GCD(a,0) = a
Time Complexity
Modulo: O(log min(a,b))
Subtraction: O(max(a,b))
Applications
Simplifying fractions, cryptography, number theory
Quick Tips
Use modulo method for large numbers
GCD of two primes is always 1
GCD(a,b) × LCM(a,b) = a × b
Algorithm works for any positive integers
Understanding the Euclidean Algorithm
What is the Euclidean Algorithm?
The Euclidean Algorithm is an ancient and elegant method for finding the Greatest Common Divisor (GCD) of two integers. Named after the ancient Greek mathematician Euclid, it's one of the oldest known algorithms and is still widely used today due to its efficiency and simplicity.
Why is it Important?
- •Efficient: Much faster than prime factorization
- •Fundamental: Used in cryptography and number theory
- •Practical: Essential for simplifying fractions
- •Mathematical: Demonstrates important mathematical principles
How It Works
Core Principle
The algorithm is based on the principle that the GCD of two numbers doesn't change if the larger number is replaced by its difference with the smaller number.
Mathematical Foundation
Key insight: GCD(a,b) = GCD(b, a mod b)
This reduces the problem size in each iteration until we reach GCD(x,0) = x
Historical Note: This algorithm appears in Euclid's "Elements" (circa 300 BCE) and is one of the earliest known algorithms still in common use today.
Real-World Applications
Cryptography
RSA encryption, key generation, and modular arithmetic operations
Mathematics
Fraction simplification, solving Diophantine equations, and modular arithmetic
Computer Science
Algorithm design, optimization problems, and computational number theory