GCD/LCM Calculator
Find the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of multiple numbers using the Euclidean algorithm with step-by-step solutions.
Enter Numbers
Enter at least 2 positive integers separated by commas
Quick Examples
About GCD & LCM
Greatest Common Divisor (GCD)
The largest positive integer that divides all given numbers without remainder.
Least Common Multiple (LCM)
The smallest positive integer that is divisible by all given numbers.
Euclidean Algorithm
Efficient method for finding GCD using repeated division and remainders.
Applications
- • Simplifying fractions
- • Solving Diophantine equations
- • Finding common denominators
- • Cryptography and number theory
Calculation Tips
- •Use prime factorization for complex numbers
- •GCD of consecutive integers is always 1
- •LCM of coprime numbers is their product
- •For fractions: use GCD to simplify, LCM for common denominators
- •Check results: GCD should divide all numbers, all numbers should divide LCM
Mathematical Background
Number Theory Concepts
- •GCD is also known as Greatest Common Factor (GCF)
- •LCM is also known as Least Common Denominator (LCD)
- •Euclidean algorithm is over 2000 years old
- •Prime factorization provides insight into number structure
Practical Applications
- •Music theory: finding harmonic intervals
- •Engineering: gear ratios and mechanical systems
- •Computer science: algorithm optimization
- •Scheduling: finding common time periods