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\begin{document}

\begin{center}
{\large{\textsc{Homework 11}}} \\
\textsc{Math 3345 -- Autumn 2022 -- Kutler}
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Please complete the following problems on your own paper. Solutions should be written clearly, legibly, and with appropriate style.

\begin{enumerate}
\item \textbf{{[Falkner Section 4 Exercise 14]}} Let $a, b, c \in \ZZ$. Prove the following statements.
\begin{enumerate}
\item $a$ divides $a$.
\item If $a$ divides $b$ and $b$ divides $a$, then $b = a$ or $b = -a$.
\item If $a$ divides $b$ and $b$ divides $c$, then $a$ divides $c$.
\end{enumerate}

\item \textbf{{[Falkner Section 5 Exercise 5]}} Prove by induction that for each $n \in \NN$, $5$ divides $7^n - 2^n$.

\item \textbf{[The sieve of Eratosthenes]}
\begin{enumerate}
\item Write all of the integers from $1$ to $100$ (perhaps as a $10 \times 10$ array), and do the following.
\begin{enumerate}
\item Cross out the number $1$.
\item Circle the number $2$, then cross out all other numbers divisible by $2$.
\item Circle the number $3$, then cross out all other numbers divisible by $3$.
\item Repeat this procedure: Circle the smallest number $d$ which is not crossed out, then cross out all other numbers divisible by $d$. Stop once every integer from $1$ to $100$ is either circled or crossed out.
\end{enumerate}

\item Explain why the circled numbers are all of the primes less than $100$.
\end{enumerate}
\end{enumerate}

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\begin{center}
\textbf{Practice Problems}
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It is strongly recommended that you complete the following problems. There is no need to write up polished, final versions of your solutions (although you may find this a useful exercise). Please do not submit any work for these problems.

\begin{enumerate}
\item Explain why $0$ is the only integer with infinitely many divisors.

\item \textbf{{[Falkner Section 4 Exercise 13]}} Let $a, b, c \in \ZZ$. Prove the following statements.
\begin{enumerate}
\item If $a | b$ and $a | c$, then $a | (b+c)$ and $a | (b - c)$.
\item If $a | b$ or $a | c$, then $a | bc$.
\item If $a | b$, then $a | (-b)$.
\item If $a | b$, then $(-a) | b$.
\end{enumerate}

\end{enumerate}




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