\documentclass[12pt]{article}
%
% Doug Nychka's LaTeX template (8/13/2004). To create a pdf file from this 
%use pdflatex 
%
% if you read one more comment read the one that starts with ***********
%
%

\nofiles % Suppress all those pesky files!

% include some color support (see examples below)
\usepackage[usenames,dvipsnames]{color}
%
% set background color 
\definecolor{Fcolor}{rgb}{.1,.1,.1} % A cool manila
\definecolor{Pcolor}{rgb}{1,1,1} % A cool manila
\pagecolor{Pcolor} 

% see the Tex files associated with dvipsnames
% for some  predefined colors.


%
% this little style file allows you to 
% step through a page adding items 
% 
\usepackage{isolatin1}
\usepackage{pause}



%
% include a graphics package for importing figures
%
\usepackage{graphicx}
%
% this nice package supports pdf, png, and  jpeg in pdflatex 
%( but just eps for latex)
% png and jpg formats for images are particularly efficient.
% In R use the pdf/png function to create a pdf/png graphics file. 
% If you have a ps file in  UNIX use   convert foo.ps foo.pdf
% to make it into a pdf version 
% 
% If the postscript file has too much white space around it 
% use ps2epsi foo.ps ; epstopdf foo.eps foo.pdf to shrink wrap the figure
%
% or use the utility xv to crop images manually. 
%
%
%******* The magic of pdflatex, 
% These options give information to the acrobat viewer for displaying. 
% Use 
% pdflatex example.tex  
% to create example.pdf for viewing  ( e.g. acroread example.pdf)
% if you comment out the next two lines you can just use latex to make the 
% usual kind of dvi or postscript output. But you will have to use 
% postscript figures instead of pdf ones.
%

\RequirePackage[pdftex,pdfpagemode=none, pdftoolbar=true,
                pdffitwindow=true,pdfcenterwindow=true]{hyperref}
%
% change these to suit! Margins assume an inch as default 
%
\setlength{\paperheight}{7in}
\setlength{\paperwidth}{9in}
\setlength{\textheight}{6.75in}
\setlength{\textwidth}{7.5in}
\setlength{\evensidemargin}{-.75in}
\setlength{\oddsidemargin}{-.75in}
\setlength{\topmargin}{-.75in}
\setlength{\parskip}{.125in}
\setlength{\parindent}{0in}

%
%


% set background color
\definecolor{Pcolor}{rgb}{1,1,1} % A rosy manila
\pagecolor{Pcolor}

% Define heading color. 
\definecolor{Hcolor}{rgb}{.3,.3,.3} % my heading color

%
% these are  my basic page headings 
% note how the color is set 
%
\def\mytitle#1{{\color{Hcolor}{\it \LARGE #1}\\ }}

\def\MYTITLE#1{{\color{Hcolor}{\huge #1}\\
                      \rule[.15in]{\textwidth}{.03in} }}
\def\myTITLE#1{{\color{Hcolor}{\huge #1}\\}}

% just to make the math work, everyone does this differently ...
\newcommand{\BLD}[1]{\mbox{\boldmath $#1$}}

% low tech way to start a new "slide" and add a background image
% This has been tuned to the aspect paper sizes declared above. 
\newcommand{\emptyBS}{
   \newpage
   \vspace*{-1in} 
   \hspace*{-.5in}
   %\includegraphics[height=\paperheight, 
   %    width=\paperwidth]{dia0231_med.jpg} % old fire
   %\vspace*{-\paperheight} 
   \vspace*{.5in} 
   %\hspace*{-\paperwidth}
}

\newcommand{\BS}{
   \newpage
   \vspace*{-1in} 
  % \hspace*{-.5in}
	{\begin{center} Euklidischer Teileralgorithmus\end{center}}
   %\includegraphics[height=\paperheight, 
   %    width=\paperwidth]{dia0230_back.jpg} % old fire
   %\vspace*{-\paperheight} 
   \vspace*{.5in} 
   %\hspace*{-\paperwidth}
}

% uncomment this version for just a new page and no background image. 
% note that you still get a background color from setting \pagecolor above
%\newcommand{\BS}{ \newpage}

\begin{document} 
\hypertarget{start}{}
 %%%%%%%%%%%
 {\color{Black} %begin default text color
 %{\color{White} %begin default text color
 {\LARGE  % fault is LARGE size font
  { \sf   % default face is sans serif
 %%%%%%%%%%%%

   %\vspace*{-1in} 
   %\hspace*{-.5in}
   %\includegraphics[height=\paperheight, 
   %    width=\paperwidth]{dia0230_back.jpg} % old fire
   %\vspace*{-\paperheight} 
   %\vspace*{.5in} 
   %\hspace*{-\paperwidth}

\BS
\MYTITLE{Euklidischer Teileralgorithmus}

Alexander \"Olzant\\
%%%%%%%%%%%%
%%%%%%%%%%%%%%%
\vspace*{.5in} % adjust this to makes things fit nicely 
%\includegraphics[height=1.5in]{dia0231_med.jpg} 
%\hfill
%\includegraphics[height=1in]{dia0231_med.jpg} 
%\\{\it \normalsize
%Supported by the National Science Foundation DMS
%}

\BS

\MYTITLE{Euklidischer Teileralgorithmus}

Ziel: gr\"osster gemeinsamen Teiler
\pause

Euklid (\~{} 300 vC)

\BS

\MYTITLE{Voraussetzungen}

\begin{enumerate} 
\item 
{f\"ur zwei ganze Zahlen gilt:} 

$$ b | a \Rightarrow ggt(a,b) = b $$

\item 
{Rekursionsschritt:}

$$ a = b t + r \Rightarrow ggt(a,b) = ggt(b,r) $$
(f\"ur $ t, r \in \!N $ )
\end{enumerate} 

\BS
\MYTITLE{Trivialfall}

wenn eine Zahl {\bf a} die andere ({\bf a}) teilt, ist 
sie der gr\"osste gemeinsame Teiler der beiden Zahlen

$$ b | a \Rightarrow ggt(a,b) = b $$

keine Zahl kann durch eine Zahl geteilt werden, die gr\"osser
ist als die Zahl selbst.

\BS
\MYTITLE{Rekursionsschritt}

da der $ggt(a,b)$ ein Vielfaches von $b$ ist, kann ich $b$ von
$a$ subtrahieren, der ggt von $(a-b)$ und $b$ ist der gleiche
wie der von $a$ und $b$. Vereinfacht: ich kann wiederholt
den Divisionsrest {\tt a \% b} von $a$ und $b$ bestimmen.
Wenn dieser 0 wird, ist $b$ der ggt der beiden Zahlen.

$$ a = b t + r \Rightarrow ggt(a,b) = ggt(b,r) $$
(f\"ur $ t, r \in \!N $ )

\BS
\MYTITLE{Algorithmus}

\begin{enumerate} 
\item wenn {\tt b > a}, dann vertausche {\tt a} und {\tt b}
\item wenn {\tt a} durch {\tt b} dividierbar ist, dann Ergebnis
{\tt ggt(a,b) == b} {\bf ENDE}
\item subtrahiere {\tt b} von {\tt a}
\item gehe zu {\bf 1.}
\end{enumerate} 

\BS
\MYTITLE{Beispiel}

\begin{enumerate} 
\item $a = 9690, b = 3825 \Rightarrow r = 2040$
\item $a = 3825, b = 2040 \Rightarrow r = 1785$
\item $a = 2040, b = 1785 \Rightarrow r = 255$
\item $a = 1785, {\bf b = 255} \Rightarrow r = 0$
\end{enumerate} 
{\tt ggt(9690,2040) == 255}

\BS
\MYTITLE{graphische \"Uberlegung}
%\includegraphics[width=\textwidth]{ggts.eps}
\includegraphics[width=\textwidth]{ggts1.jpg}

\BS
\MYTITLE{graphische \"Uberlegung}
%\includegraphics[width=\textwidth]{ggts.eps}
\includegraphics[width=\textwidth]{ggts2.jpg}

%\BS
%\MYTITLE{graphische \"Uberlegung}
%\includegraphics[width=\textwidth]{ggts.eps}
\includegraphics[width=\textwidth]{ggts3.jpg}

\BS
\MYTITLE{Schreibweise}
\begin{tabular}[l]{l@{\hspace{1cm}}l}
$b|a$       & b teilt a ({\tt a \% b == 0}) \\
$ggt(a,b)$  & gr\"osster gemeinsamer Teiler
\end{tabular}

%% example of target from a link
%% use extra arrows in veiwer to return


}% end LARGE font block
} % end of bold face block
} % end default text color block 
\end{document}

