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\textbf{Name:} \smallskip  \\
\textbf{Collaborators:} \smallskip \\ %%% List anyone with whom you discussed the problems
\textbf{Outside resources:} \smallskip \\ %%% List all resources used OTHER than the textbook, lecture notes, quizzes, worksheets, and previous homework assignments.
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\begin{center}
{
Math 2106, Foundations of Mathematical Proof \\
HW 2 --- Due January 30, 2017 (Mon)
}
\end{center}


\noindent Turn in the following problems from Hammack's book: \\
Chapter 4, problems 4, 20, 26\\
Chapter 5, problems 6, 8, 12, 18, 20, 24, 28 \\
Chapter 6, problems 4, 6, 8, 10, 16, 18\\
Chapter 7, problem 28\\

\noindent{\bf Additional exercises} (to be turned in)

\begin{enumerate}

\item Let $a \in \ZZ$.  Prove that $3 \mid a^2$ if and only if $3 \mid a$. (You may use the fact that every integer can be written as exactly one of $3k$, $3k + 1$, or $3k+2$ for some integer $k$.)


\item Let $a$, $b$, $c$, $d$, $x$ and $y$ be integers with $a \neq 0$ and $b \neq 0$.
\begin{enumerate}
\item If $a \mid c$, then $a^2 \mid c^2$.
\item If $a \mid c$ and $b \mid d$, then $ab \mid cd$.
\item If $a \mid c$ and $a \mid d$, then $a \mid cx + dy$.
\item If $a \mid b$ and $b \mid a$, then $a = b$ or $a = -b$.
\item If $a \nmid cd$, then $a \nmid c$ and $a \nmid d$.
\end{enumerate}

\item Prove or disprove.
\begin{enumerate}
\item The sum of two rational numbers is rational.
\item The sum of an irrational number and a rational number is
irrational.
\item The sum of two irrational numbers is irrational.
\item The product of two rational numbers is rational.
\item The product of an irrational number and a rational number is
irrational.
\item The product of two irrational numbers is irrational.
\end{enumerate}


\item Let $a$ be an irrational number, $s$ a real number, and $r$ 
a nonzero rational number.
Prove that $ar + s$ or $ar - s$ is irrational.


\end{enumerate}
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