Valentin Brandl
parent
6555d61fb7
commit
f5a3d1719e
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school/di-ma/uebung/09/09_1.jpg
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school/di-ma/uebung/09/09_1.jpg
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68
school/di-ma/uebung/09/09_1.tex
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68
school/di-ma/uebung/09/09_1.tex
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\documentclass[12pt,a4paper,german]{article}
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\usepackage{url}
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%\usepackage{graphics}
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\usepackage{times}
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\usepackage[T1]{fontenc}
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\usepackage{ngerman}
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\usepackage{float}
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\usepackage{diagbox}
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\usepackage[utf8]{inputenc}
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\usepackage{geometry}
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\usepackage{cancel}
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\usepackage{wasysym}
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\usepackage{csquotes}
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\usepackage{graphicx}
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\usepackage{epsfig}
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\usepackage{paralist}
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\usepackage{tikz}
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\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
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%%%%%%%%%% Fill out the the definitions %%%%%%%%%
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\def \name {Valentin Brandl} %
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\def \matrikel {108018274494} %
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\def \pname {Marvin Herrmann} %
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\def \pmatrikel {108018265436} %
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\def \gruppe {2 (Mi 10-12 Andre)}
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\def \qname {Pascal Brackmann}
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\def \qmatrikel {108017113834} %
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\def \uebung {9} %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DO NOT MODIFY THIS HEADER
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\newcommand{\hwsol}{
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\vspace*{-2cm}
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\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
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\noindent \pmatrikel \quad \pname \\
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\noindent \qmatrikel \quad \qname \\
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\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
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}
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\begin{document}
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%Import header
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\hwsol
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\section*{Aufgabe 9.1}
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\begin{enumerate}[1.]
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\item
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\includegraphics[width=\textwidth]{build/school/di-ma/uebung/09/09_1.jpg}
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\item $M_{perfekt} = \{\{1,5\}, \{2,7\}, \{3,8\}, \{4,6\}\}$
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Es gibt kein weiteres perfektes Matching
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$5$ und $8$ haben Grad 1, daher müssen die Kanten $\{1,5\}$ und $\{3,8\}$ im perfekten Matching vorkommen.
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$\{1,6\}$ und $\{3,6\}$ können nicht mehr verwendet werden, da die Knoten $1$ und $3$ schon im Matching
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vorkommen, also ist die einzige Kante, die von $6$ ausgehend im Matching vorkommt, $\{4,6\}$ sein.
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$1,3,4,5,6,8$ kommen schon im Matching vor, es fehlen also nur noch $2$ und $7$ und damit die Kante $\{2,7\}$
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\item $M_{maximal \land \lnot maximum} = \{\{1,4\}, \{2,7\}, \{3,6\}\}$
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\end{enumerate}
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\end{document}
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70
school/di-ma/uebung/09/09_2.tex
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school/di-ma/uebung/09/09_2.tex
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\documentclass[12pt,a4paper,german]{article}
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\usepackage{url}
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%\usepackage{graphics}
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\usepackage{times}
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\usepackage[T1]{fontenc}
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\usepackage{ngerman}
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\usepackage{float}
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\usepackage{diagbox}
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\usepackage[utf8]{inputenc}
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\usepackage{geometry}
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\usepackage{cancel}
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\usepackage{wasysym}
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\usepackage{csquotes}
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\usepackage{graphicx}
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\usepackage{epsfig}
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\usepackage{paralist}
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\usepackage{tikz}
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\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
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%%%%%%%%%% Fill out the the definitions %%%%%%%%%
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\def \name {Valentin Brandl} %
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\def \matrikel {108018274494} %
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\def \pname {Marvin Herrmann} %
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\def \pmatrikel {108018265436} %
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\def \gruppe {2 (Mi 10-12 Andre)}
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\def \qname {Pascal Brackmann}
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\def \qmatrikel {108017113834} %
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\def \uebung {9} %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DO NOT MODIFY THIS HEADER
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\newcommand{\hwsol}{
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\vspace*{-2cm}
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\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
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\noindent \pmatrikel \quad \pname \\
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\noindent \qmatrikel \quad \qname \\
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\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
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}
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\begin{document}
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%Import header
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\hwsol
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\section*{Aufgabe 9.2}
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\begin{enumerate}[1.]
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\item Ein $K_n$ mit $n \equiv 0 \mod 2$ ($n$ ist gerade) hat 2 perfekte Matchings, die jeweils versetzt jede zweite
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Kante des Kreises beinhalten
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Für ein perfektes Matching muss gelten $|M| = \frac{|V|}{2}$. Für einen Kreis $K_n$ mit $n \equiv 1 \mod 2$ ($n$
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ist ungerade) gilt $|V| = n$. Da $n$ ungerade existiert keine ganze Zahl $\frac{n}{2}$ $\Rightarrow$ es
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existiert kein perfektes Matching.
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\item
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\begin{enumerate}[i)]
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\item $M_1 = \{ \{000,100\}, \{001,101\}, \{010,110\}, \{011,111\} \}$
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\item $M_2 = \{ \{000,001\}, \{100,101\}, \{010,011\}, \{110,111\} \}$
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\item $M_3 = \{ \{000,010\}, \{001,011\}, \{100,110\}, \{101,111\} \}$
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\end{enumerate}
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Es gibt 3 perfekte Matchings.
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\item $\{ \{000,100\}, \{001,101\}, \{110,111\}, \{010,011\} \}$
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\end{enumerate}
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\end{document}
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79
school/di-ma/uebung/09/09_3.tex
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79
school/di-ma/uebung/09/09_3.tex
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\documentclass[12pt,a4paper,german]{article}
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\usepackage{url}
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%\usepackage{graphics}
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\usepackage{times}
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\usepackage[T1]{fontenc}
|
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\usepackage{ngerman}
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||||
\usepackage{float}
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\usepackage{diagbox}
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\usepackage[utf8]{inputenc}
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\usepackage{geometry}
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\usepackage{cancel}
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\usepackage{wasysym}
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\usepackage{csquotes}
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\usepackage{graphicx}
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\usepackage{epsfig}
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\usepackage{paralist}
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\usepackage{tikz}
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\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
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%%%%%%%%%% Fill out the the definitions %%%%%%%%%
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\def \name {Valentin Brandl} %
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\def \matrikel {108018274494} %
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\def \pname {Marvin Herrmann} %
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\def \pmatrikel {108018265436} %
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\def \gruppe {2 (Mi 10-12 Andre)}
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\def \qname {Pascal Brackmann}
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\def \qmatrikel {108017113834} %
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\def \uebung {9} %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DO NOT MODIFY THIS HEADER
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\newcommand{\hwsol}{
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\vspace*{-2cm}
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\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
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\noindent \pmatrikel \quad \pname \\
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\noindent \qmatrikel \quad \qname \\
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\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
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}
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\begin{document}
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%Import header
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\hwsol
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\section*{Aufgabe 9.3}
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\begin{enumerate}[1.]
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\item
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\begin{tabular}{|c|c|c|}
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\hline
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$eea(a,b)$ & $(d',x',y')$ & $(d,x,y)$ \\\hline\hline
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$eea(84,15)$ & $(3,-1,2)$ & $(3,2,-11)$ \\\hline
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$eea(15,9)$ & $(3,1,-1)$ & $(3,-1,2)$ \\\hline
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$eea(9,6)$ & $(3,0,1)$ & $(3,1,-1)$ \\\hline
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$eea(6,3)$ & $(3,1,0)$ & $(3,0,1)$ \\\hline
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$eea(3,0)$ & & $(3,1,0)$ \\\hline
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\end{tabular}
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$ggt(84,15) = 84 * 2 + 15 * (-11) = 3$
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\item
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\begin{tabular}{|c|c|c|}
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\hline
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$eea(a,b)$ & $(d',x',y')$ & $(d,x,y)$ \\\hline\hline
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$eea(93,32)$ & $(1,10,-11)$ & $(1,-11,32)$ \\\hline
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$eea(32,29)$ & $(1,-1,10)$ & $(1,10,-11)$ \\\hline
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$eea(29,3)$ & $(1,1,-1)$ & $(1,-1,10)$ \\\hline
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$eea(3,2)$ & $(1,0,1)$ & $(1,1,-1)$ \\\hline
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$eea(2,1)$ & $(1,1,0)$ & $(1,0,1)$ \\\hline
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$eea(1,0)$ & & $(1,1,0)$ \\\hline
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\end{tabular}
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$ggt(93,32) = 93 * (-11) + 32 * 32 = 1$
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\end{enumerate}
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\end{document}
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75
school/di-ma/uebung/09/09_4.tex
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75
school/di-ma/uebung/09/09_4.tex
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\documentclass[12pt,a4paper,german]{article}
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\usepackage{url}
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%\usepackage{graphics}
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\usepackage{times}
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\usepackage[T1]{fontenc}
|
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\usepackage{ngerman}
|
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\usepackage{float}
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\usepackage{diagbox}
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\usepackage[utf8]{inputenc}
|
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\usepackage{geometry}
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\usepackage{amsfonts}
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\usepackage{amsmath}
|
||||
\usepackage{cancel}
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\usepackage{wasysym}
|
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\usepackage{csquotes}
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\usepackage{graphicx}
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\usepackage{epsfig}
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\usepackage{paralist}
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\usepackage{tikz}
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\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
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%%%%%%%%%% Fill out the the definitions %%%%%%%%%
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\def \name {Valentin Brandl} %
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\def \matrikel {108018274494} %
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\def \pname {Marvin Herrmann} %
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\def \pmatrikel {108018265436} %
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\def \gruppe {2 (Mi 10-12 Andre)}
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\def \qname {Pascal Brackmann}
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\def \qmatrikel {108017113834} %
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\def \uebung {9} %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DO NOT MODIFY THIS HEADER
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\newcommand{\hwsol}{
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\vspace*{-2cm}
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\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
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\noindent \pmatrikel \quad \pname \\
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\noindent \qmatrikel \quad \qname \\
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\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
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}
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\begin{document}
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%Import header
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\hwsol
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\section*{Aufgabe 9.4}
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\begin{itemize}[1.]
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\item
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\begin{align*}
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15 * x &\equiv 6 &\mod 84 \\
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\\
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4 &= 2 * 84 - 11 * 15 \\
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1 &= 2 * 28 - 11 * 5 \\
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\Rightarrow x &= (-11) * 2 &\mod 84 \\
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x &= (-11) * 2 + j * 28 \text{ mit } j \in \mathbb{Z} \\
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\\
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15 * 6 = 90 &\equiv 6 &\mod 84
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\end{align*}
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\item
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\begin{align*}
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32 * x &\equiv 2 &\mod 93 \\
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\\
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1 &= 32 * 32 - 11 * 93 \\
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\Rightarrow x &= 32 * 2 &\mod 93 \\
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x &= 32 * 2 + j * 93 \text{ mit } j \in \mathbb{Z} \\
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\\
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32 * (-29) = -928 &\equiv 2 &\mod 93
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\end{align*}
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\end{itemize}
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\end{document}
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