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finished hw8
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hw/hw8/hw8.aux
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\@writefile{toc}{\contentsline {section}{Unit 4.1}{2}{}\protected@file@percent }
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\gdef \@abspage@last{8}
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# Fdb version 4
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|
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|
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|
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\newcommand{\union}{\cup}
|
||||
\newcommand{\intersection}{\cap}
|
||||
|
||||
\newcommand{\AllIntegers}{\mathbb{Z}}
|
||||
\newcommand{\AllNaturals}{\mathbb{N}}
|
||||
\newcommand{\AllRationals}{\mathbb{Q}}
|
||||
\newcommand{\AllReals}{\mathbb{R}}
|
||||
\newcommand{\AllComplexes}{\mathbb{C}}
|
||||
|
||||
\renewcommand{\mod}{\textbf{ mod }}
|
||||
\renewcommand{\pmod}[1]{\textbf{ (mod }#1)}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
|
||||
\pagebreak
|
||||
|
||||
\unit{4.1}
|
||||
\prob{10}
|
||||
Prove that if \(a\) and \(b\) are nonzero integers, \(a | b\), and \(a + b\) is odd, then \(a\) is odd.
|
||||
\begin{proof}
|
||||
Direct proof. Assume \(a\) and \(b\) are nonzero integers, \(a | b\), and \(a + b\) is odd. Then, by definition of divisibility, \(\exists k \in \AllIntegers, ak = b\). We also know that \(a + b\) is odd, so \(\exists j \in \AllIntegers, 2j + 1 = a + b\). Substituting \(b\) for \(ak\), we get \(2j + 1 = a + ak\). Factoring out \(a\), we get \(2j + 1 = a(1 + k)\). Since \(1 + k\) is an integer, we know that \(a\) is odd.
|
||||
\end{proof}
|
||||
\pagebreak
|
||||
\prob{26}
|
||||
Evaluate these quantites:
|
||||
\begin{align*}
|
||||
\end{align*}
|
||||
\begin{enumerate}[a)]
|
||||
\item \(-17 \textbf{ mod } 2\) = \hspace{12pt} 1
|
||||
\item \(144 \textbf{ mod } 7\) = \hspace{14pt} 4
|
||||
\item \(-101 \textbf{ mod } 13\) = \hspace{5pt}3
|
||||
\item \(199 \textbf{ mod } 19\) = \hspace{9pt} 9
|
||||
\end{enumerate}
|
||||
\pagebreak
|
||||
\prob{34}
|
||||
Decide whether each of these integers is congruent to 3 modulo 7.
|
||||
\begin{enumerate}[a)]
|
||||
\item 37
|
||||
\begin{align*}
|
||||
7 \nmid (37 - 3)\\
|
||||
37 \not\equiv 3 \pmod{7}
|
||||
\end{align*}
|
||||
\item 66
|
||||
\begin{align*}
|
||||
7 \mid (66 - 3)\\
|
||||
66 \equiv 3 \pmod{7}
|
||||
\end{align*}
|
||||
\item -17
|
||||
\begin{align*}
|
||||
7 \nmid (-17 - 3)\\
|
||||
-17 \not\equiv 3 \pmod{7}
|
||||
\end{align*}
|
||||
\item -67
|
||||
\begin{align*}
|
||||
7 \mid (-67 - 3)\\
|
||||
-67 \equiv 3 \pmod{7}
|
||||
\end{align*}
|
||||
\end{enumerate}
|
||||
|
||||
\pagebreak
|
||||
\prob{36}
|
||||
Find each of these values.
|
||||
|
||||
\textbf{Note: } The following rules exist for modular arithmetic where \(a, b, n \in \AllIntegers, n \geq 2\):
|
||||
\begin{align*}
|
||||
(a + b) \textbf{ mod } n &= ((a \textbf{ mod } n) + (b \textbf{ mod } n)) \textbf{ mod } n & \text{Addition Law}\\
|
||||
(a \cdot b) \textbf{ mod } n &= ((a \textbf{ mod } n) \cdot (b \textbf{ mod } n)) \textbf{ mod } n & \text{Multiplication Law}\\
|
||||
\end{align*}
|
||||
|
||||
\begin{enumerate}[a)]
|
||||
\item \((177 \mod 31 + 270 \mod 31) \mod 31\)
|
||||
\begin{align*}
|
||||
(177 \mod 31 + 270 \mod 31) \mod 31 &= (177 + 270) \mod 31\\
|
||||
&= 447 \mod 31\\
|
||||
&= 13
|
||||
\end{align*}
|
||||
\item \((177 \mod 31 \cdot 270 \mod 31) \mod 31\)
|
||||
\begin{align*}
|
||||
(177 \mod 31 \cdot 270 \mod 31) \mod 31 &= (177 \cdot 270) \mod 31\\
|
||||
&= 47790 \mod 31\\
|
||||
&= 19
|
||||
\end{align*}
|
||||
\end{enumerate}
|
||||
\pagebreak
|
||||
\unit{6.1}
|
||||
\prob{8}
|
||||
How many different three-letter initials with none of the letters repeated can people have?
|
||||
|
||||
\begin{align*}
|
||||
26 \cdot 25 \cdot 24 = 15600
|
||||
\end{align*}
|
||||
|
||||
\pagebreak
|
||||
|
||||
\prob{22}
|
||||
How many positive integers less than 1000,
|
||||
|
||||
\begin{enumerate}[a)]
|
||||
\item are divisible by 7?
|
||||
\begin{align*}
|
||||
\lfloor \frac{1000}{7} \rfloor = 142
|
||||
\end{align*}
|
||||
\item are divisible by 7 but not by 11?
|
||||
\begin{align*}
|
||||
\lfloor \frac{1000}{7} \rfloor - \lfloor \frac{1000}{77} \rfloor = 142 - 12 = 130
|
||||
\end{align*}
|
||||
\item are divisible by both 7 and 11?
|
||||
\begin{align*}
|
||||
\lfloor \frac{1000}{77} \rfloor = 12
|
||||
\end{align*}
|
||||
\item are divisible by either 7 or 11?
|
||||
\begin{align*}
|
||||
\lfloor \frac{1000}{7} \rfloor + \lfloor \frac{1000}{11} \rfloor - \lfloor \frac{1000}{77} \rfloor = 142 + 90 - 12 = 220
|
||||
\end{align*}
|
||||
\item are divisible by exactly one of 7 and 11?
|
||||
\begin{align*}
|
||||
\lfloor \frac{1000}{7} \rfloor + \lfloor \frac{1000}{11} \rfloor - 2 \lfloor \frac{1000}{77} \rfloor = 142 + 90 - 24 = 208
|
||||
\end{align*}
|
||||
\item are divisible by neither 7 nor 11?
|
||||
\begin{align}
|
||||
999 - 220 = 779
|
||||
\end{align}
|
||||
\item have distinct digits?
|
||||
\begin{align*}
|
||||
9 && \text{One digit}\\
|
||||
(10-1) \cdot 9 && \text{Two digits}\\
|
||||
(10-1) \cdot 9 \cdot 8 && \text{Three digits}\\
|
||||
9 + 9 \cdot 9 + 9 \cdot 8 \cdot 7 = 738 && \text{Total}
|
||||
\end{align*}
|
||||
\item have distinct digits and are even
|
||||
\begin{align*}
|
||||
4 && \text{One digit}\\
|
||||
9 + 8 \cdot 4 && \text{Two digits}\\
|
||||
9 \cdot 8 + 8 \cdot 8 \cdot 4 && \text{Three digits}\\
|
||||
4 + (9 + 8 \cdot 4) + (9 \cdot 8 + 8 \cdot 8 \cdot 4) = 373 && \text{Total}
|
||||
\end{align*}
|
||||
|
||||
\end{enumerate}
|
||||
\pagebreak
|
||||
\prob{28}
|
||||
How many license plates can be made using either three digits followed by three uppercase English letters or three uppercase English letters followed by three digits?
|
||||
|
||||
\begin{align*}
|
||||
10^3 \cdot 26^3 + 26^3 \cdot 10^3 = 3.5152 \cdot 10^7
|
||||
\end{align*}
|
||||
\pagebreak
|
||||
\end{document}
|
||||
Loading…
Reference in New Issue
Block a user