Number theory Divisibility and primes Theorem

Bézout's identity

Statement

Let $a$ and $b$ be integers, and let $d = \gcd(a, b)$ be their greatest common divisor. Then there are integers $x$ and $y$ such that $a x + b y = d$ .

Remarks

The integers $x$ and $y$ are not unique. If $(x, y)$ is a solution, then $(x + k b / d, \; y - k a / d)$ is also a solution for each integer $k$ , when $d \neq 0$ . The set of all integers $a x + b y$ is the set of the multiples of $d$ . The extended Euclidean algorithm computes one pair $(x, y)$ . For example, $\gcd(12, 42) = 6$ and $12 \cdot (-3) + 42 \cdot 1 = 6$ .

Reading path

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Divisibility definition
Greatest common divisor definition
Bézout's identity statement

Where this leads

3 pages build on this one.

Tag	`0DQVHMYAGA`
URI	`m2://0DQVHMYAGA`
Aliases	`bezout-lemma`, `bezouts-lemma`
Kind	statement, theorem
Writing	W1 stated
MSC	11A05
Status	This copy of the site has no build data. Run `m2 status 0DQVHMYAGA`.
Uses	Greatest common divisor
Same as	Wikidata Q513028 Wikipedia Bézout's identity

Bézout's identity

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

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