100 lines
3.0 KiB
TeX
100 lines
3.0 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 {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 10.3}
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\begin{enumerate}[1.]
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\item
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\begin{align*}
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29^n \mod 10
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\end{align*}
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Primzahlen sind nur duchr 1 und sich selbst zeilbar
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$29^n \equiv 9^n \mod 10$
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\begin{align*}
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9^1 &= 9 \\
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9^2 &= 81 \\
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9^3 &= 729 \\
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9^4 &= 6561 \\
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9^5 &= 59049 \\
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9^6 &= 531441
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\end{align*}
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$\Rightarrow$ $9*9$ ergibt Zahl mit $1$ am Ende, $9*1$ ergibt Zahl mit $9$ am Ende.
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$\Rightarrow$ Alle Potenzten von $9$ haben am Ende eine $9$ oder $1$ stehen
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$\Rightarrow$ ist $n$ gerade $\Rightarrow$ Dez. Darstellung von $29^n$ endet auf $1$
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$\Rightarrow$ ist $n$ ungerade $\Rightarrow$ Dez. Darstellung von $29^n$ endet auf $9$
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\item
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\begin{align*}
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11^{21} &\mod 100 \\
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\equiv (\prod\limits_{i = 1}^{10} 11^2) * 11 &\mod 100 \\
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\equiv (\prod\limits_{i = 1}^{10} 121) * 11 &\mod 100 \\
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\equiv (\prod\limits_{i = 1}^{10} 21) * 11 &\mod 100 \\
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\equiv (\prod\limits_{i = 1}^{5} 21^2) * 11 &\mod 100 \\
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\equiv (\prod\limits_{i = 1}^{5} 441) * 11 &\mod 100 \\
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\equiv (\prod\limits_{i = 1}^{5} 41) * 11 &\mod 100 \\
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\equiv (\prod\limits_{i = 1}^{2} 41^2) * 41 * 11 &\mod 100 \\
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\equiv (\prod\limits_{i = 1}^{2} 1681) * 41 * 11 &\mod 100 \\
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\equiv (\prod\limits_{i = 1}^{2} 81) * 41 * 11 &\mod 100 \\
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\equiv 81 * 81 * 41 * 11 &\mod 100 \\
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\equiv 6561 * 41 * 11 &\mod 100 \\
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\equiv 61 * 41 * 11 &\mod 100 \\
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\equiv 2501 * 11 &\mod 100 \\
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\equiv 1 * 11 &\mod 100 \\
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\equiv 11 &\mod 100
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\end{align*}
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Die letzten beiden Dezimalstellen von $11^{21}$ sind $11$.
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\end{enumerate}
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\end{document}
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