The Greatest Common Divisor (GCD) algorithm for integers is a deterministic computational procedure rooted in Euclidean number theory that determines the maximal integer dividing two or more non-zero integers without remainder. This concept relies on formal definitions of divisibility, associates, and invariant reduction principles, functioning as a fundamental mechanism within algebraic number theory to simplify rational functions and solve linear Diophantine equations via Bézout's identity. As an elementary yet critical subfield operation, it establishes the theoretical foundation for prime factorization uniqueness and modular arithmetic properties essential in advanced computational mathematics.
This lecture develops core results in elementary number theory concerning divisibility, the greatest common divisor (GCD), and linear combinations of integers. It establishes that any common divisor …