114 lines
2.4 KiB
TeX
114 lines
2.4 KiB
TeX
\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 {10} %
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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 10.4}
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Gegeben:
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\begin{itemize}
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\item Anzahl Klausuren $x \leq 400$
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\end{itemize}
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\begin{align*}
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x &\equiv 4 &\mod 13 \\
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x &\equiv 7 &\mod 11 \\
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x &\equiv 3 &\mod 9
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\end{align*}
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\begin{align*}
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a_1 = 4 \text{ , } m_1 = 13 \\
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a_2 = 7 \text{ , } m_2 = 11 \\
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a_3 = 3 \text{ , } m_3 = 9 \\
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\end{align*}
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\begin{align*}
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m &= 13 * 11 * 9 = 1287 \\
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\end{align*}
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\begin{align*}
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M_1 &= \frac{m}{m_1} = 99 \\
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M_2 &= \frac{m}{m_2} = 117 \\
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M_3 &= \frac{m}{m_3} = 143 \\
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\end{align*}
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\begin{align*}
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M_1' &= 8 \\
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M_2' &= 7 \\
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M_3' &= 8 \\
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\end{align*}
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\begin{align*}
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x_1 * 99 &\equiv 1 &\mod 13 \\
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x_2 * 117 &\equiv 1 &\mod 11 \\
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x_3 * 143 &\equiv 1 &\mod 9 \\
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\Rightarrow \\
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x_1 &= 5 \\
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x_2 &= 8 \\
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x_3 &= 8 \\
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\end{align*}
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\begin{align*}
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u_1 &= x_1 * M_1 = 495 \\
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u_2 &= x_2 * M_2 = 936 \\
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u_3 &= x_3 * M_3 = 1144 \\
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\end{align*}
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\begin{align*}
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x &= 4 * 495 + 7 * 936 + 3 * 1144 &\mod 1287 \\
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x &= 11964 \equiv 381 &\mod 1287
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\end{align*}
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Test:
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\begin{align*}
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381 &\equiv 4 &\mod 13 \\
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381 &\equiv 7 &\mod 11 \\
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381 &\equiv 3 &\mod 9 \\
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381 &\leq 400
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\end{align*}
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Es wurden 381 Klausuren geschrieben.
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\end{document}
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