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\surname{BEAUCHAMP}

\vol{30}
\pyear{2006}

\adres{%  = address
Sophonsiba BEAUCHAMP\\
Department of Philomathy\\
Faulkner University\\
Yoknapatawpha\\
e-mail:sibbey@faulkner.edu
}


\received{\today}
\freceived{ }

\title{Divisors of even numbers}

\author{Sophonsiba Beauchamp}

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2000 {\itshape AMS Mathematics Subject Classification:} 53A, 53C.}

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\begin{document}
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\begin{abstract}
The greatest common divisor of the even numbers is $2$.
 \keywords{greatest common divisor, even number}
\end{abstract}

\section{Introduction}

It is well known that, while only $0$ is divisible by $0$, every
number is divisible by $1$.  We show that
every even number is divisible by $2$, but by no greater number. 

\section{Preliminary results}

Let us recall:

\begin{definition}\label{defn:even}
  A number is \textbf{even} if it is the sum of two equal numbers:
  Thus $a$ is even, if $a=b+c$, where $b=c$.
\end{definition}

\begin{lem}\label{lem:2}
  Even numbers are divisible by $2$.
\end{lem}

\begin{proof}
  Let $a$ be an even number.  Then, by Definition~\ref{defn:even},
  $a=b+c$, where $b=c$.  Thus $a=b+b=1\cdot b+1\cdot b=(1+1)\cdot
  b=2\cdot b$.
\end{proof}

\section{The main theorem}

Without further ado, let us proceed:

\begin{theorem}
  The even numbers have a greatest common divisor, which is $2$.
\end{theorem}

\begin{proof}
  We have already established in Lemma~\ref{lem:2} that $2$ divides
  every even number.  Suppose $a>2$.  We must show that there is an
  even number $b$ that is indivisible by $a$.  Let $b=2$.  Then the
  remainder of $b$ after division by $a$ is $b$ itself, namely $2$.
  But $2\neq0$.  Therefore $a$ does not divide $b$.  But $b$ was
  an even number.  Therefore not every even number is
  divisible by $a$.  But $a$ was an arbitrary number greater than
  $2$.  Therefore the theorem is proved.
\end{proof}

We hope to prove in a forthcoming paper \cite{1} that the greatest common
divisor of the square numbers is $1$.  This is believed to have been
known to the ancients \cite{2}, but the proof has been lost.

\begin{thebibliography}{99}

\bibitem {1} 
S. Beauchamp, Divisors of squares, preprint.

\bibitem {2}

Philomathus of Ancyra, \emph{Thesaurus of Mathematics,} lost manuscript.

\end{thebibliography}

\begin{addres}

\end{addres}


\end{document}


