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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 {9}								%
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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 9.4}

\begin{itemize}[1.]

    \item
        \begin{align*}
            15 * x &\equiv 6 &\mod 84 \\
            \\
            4 &= 2 * 84 - 11 * 15 \\
            1 &= 2 * 28 - 11 * 5 \\
            \Rightarrow x &= (-11) * 2 &\mod 84 \\
            x &= (-11) * 2 + j * 28 \text{ mit } j \in \mathbb{Z} \\
            \\
            15 * 6 = 90 &\equiv 6 &\mod 84
        \end{align*}

    \item
        \begin{align*}
            32 * x &\equiv 2 &\mod 93 \\
            \\
            1 &= 32 * 32 - 11 * 93 \\
            \Rightarrow x &= 32 * 2 &\mod 93 \\
            x &= 32 * 2 + j * 93 \text{ mit } j \in \mathbb{Z} \\
            \\
            32 * (-29) = -928 &\equiv 2 &\mod 93
        \end{align*}

\end{itemize}

\end{document}
