1 files changed, 62 insertions, 0 deletions
diff --git a/vorlesungen/slides/4/ggt.tex b/vorlesungen/slides/4/ggt.tex
new file mode 100644
index 0000000..77b2a1d
--- /dev/null
+++ b/vorlesungen/slides/4/ggt.tex
@@ -0,0 +1,62 @@
+%
+% ggt.tex -- GGT, Definition und Algorithmus
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschuöe
+%
+\begin{frame}[t]
+\frametitle{Grösster gemeinsamer Teiler}
+\vspace{-15pt}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Gegeben: $a,b\in\mathbb Z$
+\\
+Gesucht: grösster gemeinsamer Teiler $\operatorname{ggT}(a,b)$
+\end{block}
+\begin{block}{Euklidischer Algorithmus}
+$a_0 = a$, $b_0=b$
+\begin{align*}
+a_0&=b_0q_0 + r_0 & a_1 &=b_0 & b_1&=r_0 \\
+a_1&=b_1q_1 + r_1 & a_2 &=b_1 & b_2&=r_1 \\
+a_2&=b_2q_2 + r_2 & a_3 &=b_2 & b_3&=r_2 \\
+   &\;\vdots      &     &     &    &     \\
+a_n&=b_nq_n + r_n & r_n &= 0  & r_{n-1}&)=\operatorname{ggT}(a,b)
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{$\operatorname{ggT}(15,25) = 5$}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.09]
+\draw[->] (-1,0) -- (65,0) coordinate[label={$a$}];
+\draw[->] (0,-1) -- (0,65) coordinate[label={right:$b$}];
+\begin{scope}
+\clip (-1,-1) rectangle (65,65);
+\foreach \x in {0,...,4}{
+	\draw[line width=0.2pt] ({\x*15},-2) -- ({\x*15},65);
+}
+\foreach \y in {0,...,2}{
+	\draw[line width=0.2pt] (-2,{\y*25}) -- (65,{\y*25});
+}
+\foreach \x in {0,5,...,120}{
+	\draw[color=blue] ({\x+2},-2) -- ({\x+2-70},{-2+70});
+	\node[color=blue] at ({0.5*\x-0.5},{0.5*\x-0.5}) [rotate=-45,above] {\tiny $\x$};
+}
+\foreach \x in {0,...,4}{
+	\foreach \y in {0,...,2}{
+		\fill[color=red] ({\x*15},{\y*25}) circle[radius=0.8];
+	}
+}
+\foreach \x in {0,5,...,60}{
+	\fill[color=blue] (\x,0) circle[radius=0.5];
+	\node at (\x,0) [below] {\tiny $\x$};
+}
+\end{scope}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}