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--- a/UE/ue2.tex +++ b/UE/ue2.tex @@ -4,13 +4,14 @@ \usepackage{fullpage} \usepackage[ngerman]{babel} -\usepackage[T1]{fontenc} -\usepackage{lmodern} +\usepackage{fixltx2e} %Deutschsprach Bugs +%\usepackage[T1]{fontenc} +%\usepackage{lmodern} \usepackage{amsthm} \usepackage{graphicx} \usepackage{fancyhdr} \usepackage{color} -\usepackage{ngerman} +%\usepackage{ngerman} \def\P{\mathbb{P}} @@ -64,6 +65,12 @@ erhält man nun durch Einsetzen der linken Seite von \eqref{gl0} in \eqref{gl1} \end{align} \end{subequations} \subsection*{8. Aufgabe} +\texttt{Man löse das folgende System von linearen Kongruenzen +\begin{align} + x \equiv 2 \mod 3, x \equiv 3 \mod 5, x \equiv 4 \mod 7 +\end{align} +einmal mit Hilfe der Formel aus dem Chinesischen-Restsatz und einmal, indem man die allgemeine Lösung der ersten Konrguenz in die zweite einsetzt, und dann die allgemeine Lösung der ersten zwei Konrguenzen in die dritte. +}\\ System von Gleichungen mit Teilerfremden $m_i$ \begin{align} x & \equiv a_i\mod m_i\\ @@ -105,7 +112,7 @@ Lösen durch einsetzen: \end{align} \subsection*{9. Aufgabe} -{\texttt{Man bestimme alle Lösungen der Kongruenz $f(x)=2x^{3}-3x^{2}+5x+6 \equiv 0 \mod 120$ durch Zurückführen auf die entsprechenden Lösungen mod 8, mod 3 und mod 5 und Anwendung des Chinesischen Restsatzes. }} \\ +\texttt{Man bestimme alle Lösungen der Kongruenz $f(x)=2x^{3}-3x^{2}+5x+6 \equiv 0 \mod 120$ durch Zurückführen auf die entsprechenden Lösungen mod 8, mod 3 und mod 5 und Anwendung des Chinesischen Restsatzes. }\\ \begin{align} f(x) &= 2x^3-3x^2+5x+6 \equiv 0 \mod 120 @@ -164,7 +171,7 @@ insbesondere erhält man daraus für die Mächtigkeiten \subsection*{12. Aufgabe} Ansatz von der Tafel: \begin{align} - \phi(p^eq^f) = (p^e-p^{e-1})(q^f-q^{f-1}) + \varphi(p^eq^f) = (p^e-p^{e-1})(q^f-q^{f-1}) \end{align} Lösgunen für $k=1,2,3,4$: \begin{enumerate} -- 2.47.3