UE1.1 + UE1.4

diff --git a/ue1.pdf b/ue1.pdf
index 84ddd906d1af8fc42d7d979368420a389cb64f30..c5a363fbd209ce6220d982e95678832928d3cd52 100644 (file)
Binary files a/ue1.pdf and b/ue1.pdf differ

diff --git a/ue1.tex b/ue1.tex
index aaca2bb99af5fd21dcd096e744dc1595b4957877..7899f324413c587f1b7b98f64c0772cb1de00620 100644 (file)
--- a/ue1.tex
+++ b/ue1.tex
@@ -1,8 +1,13 @@
-\documentclass[a4paper,10pt]{article}
+\documentclass[a4paper,10pt,fleqn]{article}
 \usepackage[utf8x]{inputenc}
 \usepackage{amsmath,amssymb}
 \usepackage{fullpage}
 
+
+\def\Z{\mathbb{Z}}
+\def\kgV{\text{kgV}}
+\def\ggT{\text{ggT}}
+\def\sgn{\text{sgn}}
 %opening
 \title{}
 \author{Peter Schaefer}
@@ -10,10 +15,19 @@
 \begin{document}
 \maketitle
 \section*{1.Übung}
-\subsection*{2. Aufgabe}
+\subsection*{1. Aufgabe}
+\begin{align}
+ S_n &= \frac 1 1 + \frac 1 2 + \cdots + \frac 1 n \in \Z\\
+ n &= 1 ;\\ & \Rightarrow S_1 \in \Z\\
+n & \geq 2 :\\
+a & = \kgV(1,2,\dots,n)\\
+S_n &= \frac{\frac a 1 + \frac a 2+ \cdots \frac a n} a\\ &= 
+\frac{ \overbrace{\frac a 1 + \frac a 2 + \cdots}^{\text{gerade}} +  \overbrace{\frac a {2^k}}^{\text{ungerade}} + \overbrace{\cdots + \frac a n}^{\text{gerade}}} {\underbrace{a}_{\text{gerade}}} = \frac{\text{ungerade}}{\text{gerade}} \Rightarrow S_n \notin \Z,n \in \{2,3,\dots\}
+\end{align}
 
-\begin{eqnarray*}
-&&\begin{array}{ccccccc}
+\subsection*{3. Aufgabe}
+\begin{align}
+&\begin{array}{ccccccc}
  r_{i-2} & r_{i-1}& q_i & x_{i-2} & x_{i-1} & y_{i-2} & y_{i-1}\\
  312 & 269 & 1 & 1 & 0 & 0 & 1\\
  269 & 43 & 6 & 0 & 1 & 1 & -1\\
@@ -21,10 +35,18 @@
  11 & 10 & 1 & -6 & 19 & 7 & - 22\\
  10 &  1 & & 19 & -25 & -22 & 29\\
 \end{array}\\
-\text{ggT}(312,269) & = & 1\\
-1 & = & -25*312+29*269\\
-5 & = & -125*312+145*269\\
-x & = & 5\cdot\binom{-25}{29}+\cdot
-\end{eqnarray*}
+\ggT(312,269) & = & 1\\
+1 & =  -25*312+29*269\\
+5 & =  -125*312+145*269\\
+x & =  5\cdot\binom{-25}{29}+\cdot
+\end{align}
+
+\subsection*{4. Aufgabe}
+\begin{align}
+ v &= \kgV(a,b) = \frac{|ab|}{\ggT(a,b)}\\
+ &= a\cdot \frac{\sgn(a) |b|}{\ggT(a,b)} = b\cdot \frac{\sgn(b) |a|}{\ggT(a,b)}\\
+&a|w \wedge b|w \Rightarrow ab|bw \wedge ab|aw \Rightarrow ab|\ggT(aw,bw)\Rightarrow|w|\ggT(a,b) \Rightarrow \frac{ab}{\ggT(a,b)}\Big||w|
+\end{align}
+
 
 \end{document}