##  [Bézout's Identity](/bezouts-identity-0) 

 Definition

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

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

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

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

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

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

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

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

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.