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\def\HW{10}
\def\DUE{12/4/2018}
\def\NAME{YOUR NAME HERE}
\title[Homework \HW]{Homework \HW\\
Math 308\\
\small Due: \DUE}
\author{}
%\date{}                                           % Activate to display a given date or no date

\begin{document}
%\maketitle %Comment out this line, and uncomment the following for your assignment
\begin{flushright}
\NAME\\
Math 308\\
Homework \HW\\
\DUE
\end{flushright}


\noindent Throughout, let $n$ be a non-zero integer, and let $a, b$, and $c$ be integers.

\begin{enumerate}[1.]
\item Find the least residue of $a \ (\mathrm{mod}\ n)$ for each of the following:\\ \smallskip
\centerline{(a) $a = 16, n = 5$;  \quad (b)  $a = 545, n = 12$; \quad 
		(c) $a = -33, n = 22$, (d) $a = 7, n = 7$.
} 
\item Show that if $a \equiv b \pmod{n}$, then $a^k \equiv b^k \pmod{n}$ for all $k \in \ZZ_{\ge 0}$. 
\item Division.
\begin{enumerate}
\item Show that if $\gcd(a,b) = 1$, then $ac \equiv bc \pmod{n}$ implies $a \equiv b \pmod{n}$. 
\item Prove or disprove: For all integers $a$ and $b$, if $ab \equiv 0 \pmod{n}$, then $a \equiv 0 \pmod{n}$ or $b \equiv 0 \pmod{n}$.
\item Prove or disprove: For all integers $n>3$ and $a, b \in \ZZ$, if $a^2 \equiv 4 \pmod{n}$, then $a \equiv 2 \pmod{n}$.
\end{enumerate}

\item Rewrite, fine-tune, and complete the book's proof of Fermat's little theorem. 

\item While it is not true that $x^m \equiv_m x$ for all $m \in \ZZ_{\ge 2}$, $x \in \ZZ$, some generalizations of Fermat's little theorem hold.

\begin{enumerate}
\item Find an example of $m \in \ZZ_{\ge 2}$ and $x \in \ZZ$ where $x^m \not\equiv_m x$.
\item Show that $x^3 \equiv_6 x$ for all $x \in \ZZ$. 
\end{enumerate}

\item Classify each of the following as finite, countably infinite, or uncountable. Justify/prove your answers.
\begin{enumerate}
\item The positive prime integers, $\{ p \in \ZZ_{>0} ~|~ p \text{ is prime }\}$.
\item The rational numbers with denominators a power of 2, $\{ n/2^m ~|~ n \in \ZZ, m \in \ZZ_{\ge 0}\}$.
\item The complex numbers, $\CC$.
\item The irrational numbers, $\RR - \QQ$. 
\end{enumerate}

\item Show that for any set $A$, we have  $|A| < |\cP(A)|$, where $\cP(A)$ is the power set of $A$.  (See class notes.)

\end{enumerate}





\end{document}