11A05-CorollaryOfBezoutsLemma.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{CorollaryOfBezoutsLemma}
\pmcreated{2013-03-22 14:48:16}
\pmmodified{2013-03-22 14:48:16}
\pmowner{pahio}{2872}
\pmmodifier{pahio}{2872}
\pmtitle{corollary of B\'ezout's lemma}
\pmrecord{15}{36459}
\pmprivacy{1}
\pmauthor{pahio}{2872}
\pmtype{Theorem}
\pmcomment{trigger rebuild}
\pmclassification{msc}{11A05}
\pmsynonym{Euclid's lemma}{CorollaryOfBezoutsLemma}
\pmsynonym{product divisible but factor coprime}{CorollaryOfBezoutsLemma}
%\pmkeywords{divisibility}
\pmrelated{GreatestCommonDivisor}
\pmrelated{DivisibilityInRings}
\pmrelated{DivisibilityByProduct}

\endmetadata

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\newtheorem*{thmplain}{Theorem}
\begin{document}
\begin{thmplain}
\;If \,$\gcd(a,\,c) = 1$\, and \,$c|ab$, \,then \,$c|b$.
\end{thmplain}
{\em Proof.} \,\PMlinkname{B\'ezout's lemma}{BezoutsLemma} gives the integers $x$ and $y$ such that 
\,$xa+yc = 1$. \,This implies that \,$xab+ybc = b$, \,and because here the both summands are divisible by $c$, so also the sum, i.e. $b$, is divisible by $c$ .

\textbf{Note.} \,A similar theorem holds in all \PMlinkname{B\'ezout domains}{BezoutDomain}, also in {\em B\'ezout rings}.
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