Bezout's lemma (number theory) BezoutsLemma 2002-05-27 16:29:04 2004-02-13 08:35:58 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $a,b$ be integers, not both zero. Then there exist two integers $x,y$ such that: $$ax+by=\gcd(a,b).$$ This does not only work on $\mathbb{Z}$ but on every integral domain where an Euclidean valuation has been defined.