even numberEvenNumber2003-09-05 15:11:232006-08-10 13:15:54Definitionodd numbereven integerodd integerevenodd% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}{\bf Definition} Suppose $k$ is an integer.
If there exists an integer $r$ such that $k=2r+1$, then $k$ is an {\bf odd number}.
If there exists an integer $r$ such that $k=2r$, then $k$ is an {\bf even number}.
The concept of even and odd numbers are most easily understood in
the binary base. Then the above definition simply \PMlinkescapetext{states} that even numbers end
with a $0$, and odd numbers end with a $1$.
\subsubsection{Properties}
\begin{enumerate}
\item Every integer is either even or \PMlinkescapetext{odd}. This can be proven
using induction, or using the fundamental theorem of arithmetic.
\item An integer $k$ is even (\PMlinkescapetext{odd}) if and only if $k^2$ is even (\PMlinkescapetext{odd}).
\end{enumerate}