Bézout's Lemma
In number theory, Bézout's Lemma, also called Bézout's Identity, states that for any integers 
and 
with greatest common denominator 
, there exist integers 
and 
such that 
. Furthermore, the integers of the form 
are exactly the multiples of . Bézout's Lemma is a foundational result in number theory that implies many other theorems, such as Euclid's Lemma and the Chinese Remainder Theorem.
To see an example of Bézout's Lemma, let and be 
and 
respectively. Note that 
, The Lemma thus states that there exist integers and such that 
. A solution 
to this equation is 
.
(Note: This article is a work in progress! I don't believe AoPS has sandboxes, sadly. This should eventually replace Bezout's Lemma as the main article).
