GCD, LCM, and Euclidean Algorithm

GCD, LCM, and Euclidean Algorithm

Greatest Common Divisor

Least Common Multiple

Relationship

Euclidean Algorithm

Extended Euclidean Algorithm

Algorithm:

1. Apply Euclidean algorithm to find quotients
2. 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