76 lines
1.9 KiB
TeX
76 lines
1.9 KiB
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 9.4}
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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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