Bézout's Identity/Proof 5
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Theorem
Let $a, b \in \Z$ such that $a$ and $b$ are not both zero.
Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.
Then:
- $\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$
That is, $\gcd \set {a, b}$ is an integer combination (or linear combination) of $a$ and $b$.
Furthermore, $\gcd \set {a, b}$ is the smallest positive integer combination of $a$ and $b$.
Proof
Let $\gcd \set {a, b} = d$.
Let $\dfrac a d = p$ and $\dfrac b d = q$.
From Integers Divided by GCD are Coprime:
- $\gcd \left\{{p, q}\right\} = 1$
From Integer Combination of Coprime Integers:
- $\exists x, y \in \Z: p x + q y = 1$
The result follows by multiplying both sides by $d$.
$\blacksquare$
Source of Name
This entry was named for Étienne Bézout.
Sources
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory