diff --git a/school/di-ma/uebung/10/10_1.tex b/school/di-ma/uebung/10/10_1.tex new file mode 100644 index 0000000..009003f --- /dev/null +++ b/school/di-ma/uebung/10/10_1.tex @@ -0,0 +1,66 @@ +\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} diff --git a/school/di-ma/uebung/10/10_2.tex b/school/di-ma/uebung/10/10_2.tex new file mode 100644 index 0000000..f0f7d0b --- /dev/null +++ b/school/di-ma/uebung/10/10_2.tex @@ -0,0 +1,101 @@ +\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} diff --git a/school/di-ma/uebung/10/10_3.tex b/school/di-ma/uebung/10/10_3.tex new file mode 100644 index 0000000..43d62ec --- /dev/null +++ b/school/di-ma/uebung/10/10_3.tex @@ -0,0 +1,99 @@ +\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} diff --git a/school/di-ma/uebung/10/10_4.tex b/school/di-ma/uebung/10/10_4.tex new file mode 100644 index 0000000..707f646 --- /dev/null +++ b/school/di-ma/uebung/10/10_4.tex @@ -0,0 +1,113 @@ +\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} diff --git a/school/di-ma/uebung/10/hausuebung10.pdf b/school/di-ma/uebung/10/hausuebung10.pdf new file mode 100644 index 0000000..8de020a Binary files /dev/null and b/school/di-ma/uebung/10/hausuebung10.pdf differ