102 lines
2.6 KiB
TeX
102 lines
2.6 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 10.2}
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\begin{enumerate}[1.]
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\item $a \equiv b (\mod m) \Rightarrow a^2 \equiv b^2 (\mod m)$
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Beweis:
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$a \equiv b \text{ mod m}$
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$\Rightarrow a = b + k\cdot m, k \in \mathbb{Z}$
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$\Rightarrow a^2 = (b+k\cdot m)^2 = b^2 + 2b\cdot k \cdot m + k^2 \cdot m^2 = b^2 + (2\cdot b \cdot k + k^2\cdot m)\cdot m$
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$\Rightarrow a^2 \equiv b^2 \text{ mod m}$
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q.e.d
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\item $a^2 \equiv b^2 (\mod m) \Rightarrow a \equiv b (\mod m)$
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Gegenbeispiel:
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a $\equiv$ 2 mod 5 und b $\equiv$ -2 $\equiv$ 3 mod 5
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Es gilt: $a^2 \equiv 2^2 \equiv 4 \equiv 3^2 \equiv b^2 \text{ mod } 5$.
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Aber $a \not \equiv b \text{ mod } 5$ und
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\item $a^2 \equiv b^2 (\mod m) \Rightarrow (a \equiv b (\mod m) \lor a \equiv -b (\mod m))$
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Gegenbeispiel a=2, b=4 und m=12
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Es gilt: $a^2 \text{ mod 12 } \equiv 2^2 \text{ mod 12 }\equiv 4 \text{ mod 12 } \equiv 4^2 \text{ mod 12 }\equiv b^2 \text{ mod 12 }$
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Aber
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$2 \not \equiv 4 \text{ mod 12}$ und
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$2 \not \equiv -4 \text{ mod 12} \equiv 8 \text{ mod 12 }$
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\item $a \equiv b (\mod m) \Rightarrow a^2 \equiv b^2 (\mod m^2)$
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Gegenbeispiel a=4 b=9 und m=5, also $m^2=25$
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$\Rightarrow a \equiv b \text{ mod 5}$
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$\Rightarrow a^2 \equiv 16 \not \equiv 81 \equiv 6 \equiv b^2 \text{ mod 25}$
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$\Rightarrow a^2 \not \equiv b^2 \text{ mod 25}$
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\end{enumerate}
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\end{document}
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