The statement that for any integers a and b not both zero there exist integers x and y such that gcd(a,b) = ax + by; that is, the greatest common divisor can be expressed as an integer linear combination of a and b.

Principle

The greatest common divisor of two integers generates the ideal they span in Z; the Euclidean algorithm produces coefficients realizing the gcd as a linear combination and organizes solvability of related Diophantine equations.

Demonstration

Concrete example: for a = 30 and b = 21, gcd(30,21) = 3 and one finds integers x = -2, y = 3 so that 3 = (-2)·30 + 3·21. The coefficients arise from back‑substitution in the Euclidean algorithm.

Misapplication

Assuming the representation is unique — coefficients x,y are not unique; assuming the identity holds in rings that are not principal ideal domains without checking that the ideal generated by a and b is principal.

Consequence

Gives a direct method to decide coprimality (gcd = 1 iff there exist x,y with ax+by=1), solves linear Diophantine equations ax+by=c when gcd divides c, and yields algorithms for modular inverses.

Reversal

If integers x,y exist with ax+by = d then d is a common divisor of a and b; when ax+by = 1 the reversal identifies a and b as coprime. The contrast highlights how existence of such a combination characterizes gcd properties.

Boundary

Classically stated for integers; it extends to principal ideal domains but fails in general rings where ideals need not be principal. It presumes integer coefficients and standard gcd theory.

Semantic Tension

Tension between the numeric notion 'greatest' (largest common divisor by size or divisibility) and the ideal‑theoretic view (generator of the ideal (a,b)): Bézout's identity ties these perspectives but in non‑PIDs they diverge.

Synthesis

Bézout's Identity unites algorithmic and structural views: the Euclidean algorithm produces explicit coefficients expressing the gcd as a linear combination, which both characterizes coprimality and provides constructive tools for solving linear Diophantine problems.