114 lines
2.4 KiB
TeX
114 lines
2.4 KiB
TeX
|
\documentclass[12pt,a4paper,german]{article}
|
||
|
|
\usepackage{url}
|
||
|
|
%\usepackage{graphics}
|
||
|
|
\usepackage{times}
|
||
|
|
\usepackage[T1]{fontenc}
|
||
|
|
\usepackage{ngerman}
|
||
|
|
\usepackage{float}
|
||
|
|
\usepackage{diagbox}
|
||
|
|
\usepackage[utf8]{inputenc}
|
||
|
|
\usepackage{geometry}
|
||
|
|
\usepackage{amsfonts}
|
||
|
|
\usepackage{amsmath}
|
||
|
|
\usepackage{cancel}
|
||
|
|
\usepackage{wasysym}
|
||
|
|
\usepackage{csquotes}
|
||
|
|
\usepackage{graphicx}
|
||
|
|
\usepackage{epsfig}
|
||
|
|
\usepackage{paralist}
|
||
|
|
\usepackage{tikz}
|
||
|
|
\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
|
||
|
|
|
||
|
|
%%%%%%%%%% Fill out the the definitions %%%%%%%%%
|
||
|
|
\def \name {Valentin Brandl} %
|
||
|
|
\def \matrikel {108018274494} %
|
||
|
|
\def \pname {Marvin Herrmann} %
|
||
|
|
\def \pmatrikel {108018265436} %
|
||
|
|
\def \gruppe {2 (Mi 10-12 Andre)}
|
||
|
|
\def \qname {Pascal Brackmann}
|
||
|
|
\def \qmatrikel {108017113834} %
|
||
|
|
\def \uebung {10} %
|
||
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||
|
|
|
||
|
|
% DO NOT MODIFY THIS HEADER
|
||
|
|
\newcommand{\hwsol}{
|
||
|
|
\vspace*{-2cm}
|
||
|
|
\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
|
||
|
|
\noindent \pmatrikel \quad \pname \\
|
||
|
|
\noindent \qmatrikel \quad \qname \\
|
||
|
|
\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
|
||
|
|
}
|
||
|
|
|
||
|
|
\begin{document}
|
||
|
|
%Import header
|
||
|
|
\hwsol
|
||
|
|
|
||
|
|
\section*{Aufgabe 10.4}
|
||
|
|
|
||
|
|
Gegeben:
|
||
|
|
\begin{itemize}
|
||
|
|
\item Anzahl Klausuren $x \leq 400$
|
||
|
|
\end{itemize}
|
||
|
|
|
||
|
|
\begin{align*}
|
||
|
|
x &\equiv 4 &\mod 13 \\
|
||
|
|
x &\equiv 7 &\mod 11 \\
|
||
|
|
x &\equiv 3 &\mod 9
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
\begin{align*}
|
||
|
|
a_1 = 4 \text{ , } m_1 = 13 \\
|
||
|
|
a_2 = 7 \text{ , } m_2 = 11 \\
|
||
|
|
a_3 = 3 \text{ , } m_3 = 9 \\
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
\begin{align*}
|
||
|
|
m &= 13 * 11 * 9 = 1287 \\
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
\begin{align*}
|
||
|
|
M_1 &= \frac{m}{m_1} = 99 \\
|
||
|
|
M_2 &= \frac{m}{m_2} = 117 \\
|
||
|
|
M_3 &= \frac{m}{m_3} = 143 \\
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
\begin{align*}
|
||
|
|
M_1' &= 8 \\
|
||
|
|
M_2' &= 7 \\
|
||
|
|
M_3' &= 8 \\
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
\begin{align*}
|
||
|
|
x_1 * 99 &\equiv 1 &\mod 13 \\
|
||
|
|
x_2 * 117 &\equiv 1 &\mod 11 \\
|
||
|
|
x_3 * 143 &\equiv 1 &\mod 9 \\
|
||
|
|
\Rightarrow \\
|
||
|
|
x_1 &= 5 \\
|
||
|
|
x_2 &= 8 \\
|
||
|
|
x_3 &= 8 \\
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
\begin{align*}
|
||
|
|
u_1 &= x_1 * M_1 = 495 \\
|
||
|
|
u_2 &= x_2 * M_2 = 936 \\
|
||
|
|
u_3 &= x_3 * M_3 = 1144 \\
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
\begin{align*}
|
||
|
|
x &= 4 * 495 + 7 * 936 + 3 * 1144 &\mod 1287 \\
|
||
|
|
x &= 11964 \equiv 381 &\mod 1287
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
Test:
|
||
|
|
|
||
|
|
\begin{align*}
|
||
|
|
381 &\equiv 4 &\mod 13 \\
|
||
|
|
381 &\equiv 7 &\mod 11 \\
|
||
|
|
381 &\equiv 3 &\mod 9 \\
|
||
|
|
381 &\leq 400
|
||
|
|
\end{align*}
|
||
|
|
|
||
|
|
Es wurden 381 Klausuren geschrieben.
|
||
|
|
|
||
|
|
\end{document}
|