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Add solution for dima 10
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Valentin Brandl 2019-01-08 15:14:11 +01:00
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66
school/di-ma/uebung/10/10_1.tex Normal file
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\documentclass[12pt,a4paper,german]{article}
\usepackage{url}
%\usepackage{graphics}
\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{ngerman}
\usepackage{float}
\usepackage{diagbox}
\usepackage[utf8]{inputenc}
\usepackage{geometry}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{cancel}
\usepackage{wasysym}
\usepackage{csquotes}
\usepackage{graphicx}
\usepackage{epsfig}
\usepackage{paralist}
\usepackage{tikz}
\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
%%%%%%%%%% 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} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% DO NOT MODIFY THIS HEADER
\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 10.1}
Kriterium:
Eine Zahl $z \in \mathbb{N}$ ist durch 37 teilbar, wenn ihre nicht-alternierende hunderter-Quersumme durch 37 teilbar ist.\\
\\
Beweis: Sei $z=a_0+a_1 \cdot 10 + a_2 \cdot 10^2 + ... + a_n \cdot 10^n$.\\
Es gilt $37 |z \Leftrightarrow z \equiv 0 \text{ mod 37 }$\\
$\Rightarrow a_0+a_1\cdot 10 + a_2 \cdot 10^2 + ... + a_n \cdot 10^n \equiv 0 \text{ mod 37}$\\
Es gilt nun $10^3 \equiv 1 \text{ mod } 37$\\
\\
$\Rightarrow a_0 + a_1 \cdot 10^1 + a_2 \cdot 10^2 + a_3 + a_4 \cdot 10 + a_5 \cdot 10^2 + ... + a_n \cdot 10^n \equiv 0 \text{ mod } 37 \Leftrightarrow 37 | z$\\
\\
Interpretiere dies als \enquote{Summe der Hunderter}:\\
$(a_0+a_1 \cdot 10 + a_2 \cdot 10^2) + (a_3 + a_4 \cdot 10 + a_5 \cdot 10^2) + ... + a_n \cdot 10^n \equiv 0 \text{ mod } 37$\\
\\
$\Rightarrow \sum_{i=0}^{\lfloor \frac{n}{3} \rfloor}
(a_{3i}+a_{3i+1}\cdot 10 + a_{3i+2} \cdot 10^2) \equiv 0 \text{ mod } 37$, aj = 0 für j>n\\
$\Rightarrow$ das vorher beschriebene Kriterium
\end{document}

101
school/di-ma/uebung/10/10_2.tex Normal file
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\documentclass[12pt,a4paper,german]{article}
\usepackage{url}
%\usepackage{graphics}
\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{ngerman}
\usepackage{float}
\usepackage{diagbox}
\usepackage[utf8]{inputenc}
\usepackage{geometry}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{cancel}
\usepackage{wasysym}
\usepackage{csquotes}
\usepackage{graphicx}
\usepackage{epsfig}
\usepackage{paralist}
\usepackage{tikz}
\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
%%%%%%%%%% 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} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% DO NOT MODIFY THIS HEADER
\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 10.2}
\begin{enumerate}[1.]
\item $a \equiv b (\mod m) \Rightarrow a^2 \equiv b^2 (\mod m)$
Beweis:
$a \equiv b \text{ mod m}$
$\Rightarrow a = b + k\cdot m, k \in \mathbb{Z}$
$\Rightarrow a^2 = (b+k\cdot m)^2 = b^2 + 2b\cdot k \cdot m + k^2 \cdot m^2 = b^2 + (2\cdot b \cdot k + k^2\cdot m)\cdot m$
$\Rightarrow a^2 \equiv b^2 \text{ mod m}$
q.e.d
\item $a^2 \equiv b^2 (\mod m) \Rightarrow a \equiv b (\mod m)$
Gegenbeispiel:
a $\equiv$ 2 mod 5 und b $\equiv$ -2 $\equiv$ 3 mod 5
Es gilt: $a^2 \equiv 2^2 \equiv 4 \equiv 3^2 \equiv b^2 \text{ mod } 5$.
Aber $a \not \equiv b \text{ mod } 5$ und
\item $a^2 \equiv b^2 (\mod m) \Rightarrow (a \equiv b (\mod m) \lor a \equiv -b (\mod m))$
Gegenbeispiel a=2, b=4 und m=12
Es gilt: $a^2 \text{ mod 12 } \equiv 2^2 \text{ mod 12 }\equiv 4 \text{ mod 12 } \equiv 4^2 \text{ mod 12 }\equiv b^2 \text{ mod 12 }$
Aber
$2 \not \equiv 4 \text{ mod 12}$ und
$2 \not \equiv -4 \text{ mod 12} \equiv 8 \text{ mod 12 }$
\item $a \equiv b (\mod m) \Rightarrow a^2 \equiv b^2 (\mod m^2)$
Gegenbeispiel a=4 b=9 und m=5, also $m^2=25$
$\Rightarrow a \equiv b \text{ mod 5}$
$\Rightarrow a^2 \equiv 16 \not \equiv 81 \equiv 6 \equiv b^2 \text{ mod 25}$
$\Rightarrow a^2 \not \equiv b^2 \text{ mod 25}$
\end{enumerate}
\end{document}

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school/di-ma/uebung/10/10_3.tex Normal file
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\documentclass[12pt,a4paper,german]{article}
\usepackage{url}
%\usepackage{graphics}
\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{ngerman}
\usepackage{float}
\usepackage{diagbox}
\usepackage[utf8]{inputenc}
\usepackage{geometry}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{cancel}
\usepackage{wasysym}
\usepackage{csquotes}
\usepackage{graphicx}
\usepackage{epsfig}
\usepackage{paralist}
\usepackage{tikz}
\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
%%%%%%%%%% 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} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% DO NOT MODIFY THIS HEADER
\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 10.3}
\begin{enumerate}[1.]
\item
\begin{align*}
29^n \mod 10
\end{align*}
Primzahlen sind nur duchr 1 und sich selbst zeilbar
$29^n \equiv 9^n \mod 10$
\begin{align*}
9^1 &= 9 \\
9^2 &= 81 \\
9^3 &= 729 \\
9^4 &= 6561 \\
9^5 &= 59049 \\
9^6 &= 531441
\end{align*}
$\Rightarrow$ $9*9$ ergibt Zahl mit $1$ am Ende, $9*1$ ergibt Zahl mit $9$ am Ende.
$\Rightarrow$ Alle Potenzten von $9$ haben am Ende eine $9$ oder $1$ stehen
$\Rightarrow$ ist $n$ gerade $\Rightarrow$ Dez. Darstellung von $29^n$ endet auf $1$
$\Rightarrow$ ist $n$ ungerade $\Rightarrow$ Dez. Darstellung von $29^n$ endet auf $9$
\item
\begin{align*}
11^{21} &\mod 100 \\
\equiv (\prod\limits_{i = 1}^{10} 11^2) * 11 &\mod 100 \\
\equiv (\prod\limits_{i = 1}^{10} 121) * 11 &\mod 100 \\
\equiv (\prod\limits_{i = 1}^{10} 21) * 11 &\mod 100 \\
\equiv (\prod\limits_{i = 1}^{5} 21^2) * 11 &\mod 100 \\
\equiv (\prod\limits_{i = 1}^{5} 441) * 11 &\mod 100 \\
\equiv (\prod\limits_{i = 1}^{5} 41) * 11 &\mod 100 \\
\equiv (\prod\limits_{i = 1}^{2} 41^2) * 41 * 11 &\mod 100 \\
\equiv (\prod\limits_{i = 1}^{2} 1681) * 41 * 11 &\mod 100 \\
\equiv (\prod\limits_{i = 1}^{2} 81) * 41 * 11 &\mod 100 \\
\equiv 81 * 81 * 41 * 11 &\mod 100 \\
\equiv 6561 * 41 * 11 &\mod 100 \\
\equiv 61 * 41 * 11 &\mod 100 \\
\equiv 2501 * 11 &\mod 100 \\
\equiv 1 * 11 &\mod 100 \\
\equiv 11 &\mod 100
\end{align*}
Die letzten beiden Dezimalstellen von $11^{21}$ sind $11$.
\end{enumerate}
\end{document}

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school/di-ma/uebung/10/10_4.tex Normal file
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\documentclass[12pt,a4paper,german]{article}
\usepackage{url}
%\usepackage{graphics}
\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{ngerman}
\usepackage{float}
\usepackage{diagbox}
\usepackage[utf8]{inputenc}
\usepackage{geometry}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{cancel}
\usepackage{wasysym}
\usepackage{csquotes}
\usepackage{graphicx}
\usepackage{epsfig}
\usepackage{paralist}
\usepackage{tikz}
\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
%%%%%%%%%% 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} %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% DO NOT MODIFY THIS HEADER
\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}

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school/di-ma/uebung/10/hausuebung10.pdf Normal file
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