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\lhead{MATH 311-02}
\chead{Problem Set \#2\answer{ solutions}}
\rhead{\answer{YOUR NAME HERE}}
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\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{{\bf due February 9, 2018}}
\begin{document}
At this point your proofs might make use of several possible
techniques: direct proof, proof by contrapositive, proof by
contradiction, and proof by construction. Prove each of the following
statements in whichever way seems to be most effective to you.
\begin{enumerate}
\item \question{For any integer $n$, if $3$ does not divide $4n^2+n$, then $3$ does not divide $n$.}
\item \question{There is a triple of integers $a$, $b$, and $c$ such
that $a^2+b^2=c^2$ (called a \emph{Pythagorean triple}) such that
$c=5$ and $a$ and $b$ are consecutive integers.}
\item \question{For any real $x$, it is the case that $x(1-x)\leq\frac14$.}
\item \question{For all integers $a$ and $b$,
$(a+b)^3\equiv (a^3+b^3)\pmod 3$.}
\item \question{For real $\alpha$ and $x$, if $\alpha$ is irrational
and $x$ is rational, then the product $\alpha x$ is irrational.}
\end{enumerate}
\end{document}