GCD, LCM, and Euclidean Algorithm
Greatest Common Divisor
Least Common Multiple
Relationship
Euclidean Algorithm
Extended Euclidean Algorithm
Algorithm:
- Apply Euclidean algorithm to find quotients
- Back-substitute to express as linear combination
Example
Find and Bézout coefficients:
Back-substitute:
So , :
Properties
Coprimality
Modular Inverse
Example
Find :
Back-substitute:
LCM
Multiple GCDs
Binary GCD (Stein's Algorithm)
Applications
- RSA key generation (computing modular inverses)
- Solving linear Diophantine equations
- Chinese Remainder Theorem
- Continued fraction expansions
- Simplifying fractions