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%%%%%%%%%% 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}								%
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\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}