112 lines
3.9 KiB
TeX
112 lines
3.9 KiB
TeX
\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 {11} %
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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 11.1}
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\begin{enumerate}[1.]
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\item Sei $p$ prim:
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\begin{align*}
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(a + b)^p &\equiv a^p + b^p &\mod p
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\end{align*}
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\begin{align*}
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(a + b)^p &\equiv \sum\limits^p_{k=0} \binom{p}{k} a^k b^{p-k} &\mod p \\
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&\equiv a^p + b^p + \sum\limits^{p-1}_{k=1} \binom{p}{k} a^k b^{p-k} &\mod p \\
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* &\equiv a^p + b^p &\mod p
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\end{align*}
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$*$: $\binom{p}{k} = p * \frac{(p-1)!}{k!(p-k)!}$
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$p$ prim $\Rightarrow$ $ggT(k!(p-k!), p) = 1$ $\Rightarrow$ $\frac{(p-1)!}{k!(p-k)!} \in \mathbb{Z}_p$
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$\Rightarrow$ $p | \binom{k}{p}$ für $1 \leq k \leq p-1$
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$\Rightarrow$ $\sum\limits_{k=1}^{p-1} \binom{p}{k} a^k b^{p-k} \equiv 0 \mod p$
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q.e.d.
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\item Sei $a,b \in \mathbb{N}, c = ggT(a,b)$
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\begin{align*}
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\varphi(a * b) &= \varphi(a) * \varphi(b) * \frac{c}{\varphi(c)}
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\end{align*}
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Zu zeigen: $\varphi(a) * \varphi(b) = \varphi(ggT(a,b)) * \varphi(kgV(a,b))$
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Seien $P'$ die gemeinsamen Primteiler von $a$ und $b$ und $A$ und $B$ die Mengen der disjunkten Primteiler von
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$a$ und $b$.
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\begin{align*}
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\varphi(a) * \varphi(b) &= \prod\limits_{p \in P'} p^{(a_p - 1)(b_p -1)} (p-1)^2 * \prod\limits_{p \in A}
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p^{(a_p-1)} (p-1) * \prod\limits_{p \in B} p^{(b_p - 1)} (p-1) \\
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&= \prod\limits_{p \in P'} p^{(min(a_p,b_p)-1)(max(a_p,b_p)-1)} (p-1)^2 * \prod\limits_{p \in A}
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p^{(a_p-1)} (p-1) * \prod\limits_{p \in B} p^{(b_p - 1)} (p-1) \\
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&= \prod\limits_{p \in P'} p^{min(a_p,b_p)-1} (p-q) * \left(
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\prod\limits_{p \in P'} p^{max(a_p,b_p)-1} (p-1) * \prod\limits_{p \in A}
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p^{(a_p-1)} (p-1) * \prod\limits_{p \in B} p^{(b_p - 1)} (p-1) \right) \\
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&= \varphi(ggT(a,b)) * \varphi(kgV(a,b)) \\
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\\
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\varphi(a * b) &= \varphi(a) * \varphi(b) * \frac{c}{\varphi(c)} \\
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&= \varphi(ggT(a,b)) * \varphi(kgV(a,b)) * \frac{ggT(a,b)}{\varphi(ggT(a,b))} \\
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&= \varphi(kgV(a,b)) * ggT(a,b) \\
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&= \prod\limits_{p \in P'} p^{max(a_p,b_p)-1} (p-1) * \prod\limits_{p \in A}
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p^{(a_p-1)} (p-1) * \prod\limits_{p \in B} p^{(b_p - 1)} (p-1) * \prod\limits_{p \in P'}
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p^{min(a_p,b_p)} \\
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&= \prod\limits_{p \in P'} p^{(a_p+b_p-1)} (p-1) * \prod\limits_{p \in A} p^{a_p-1} (p-1) *
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\prod\limits_{p \in B} p^{b_p-1} (p-1) \\
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&= \varphi(a*b)
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\end{align*}
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q.e.d.
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\end{enumerate}
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\end{document}
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