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\documentclass{math215}

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\doclabel{Math 215: Homework 5}
\docauthor{Your name here.}
\docdate{February 22, 2013}

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\begin{document}

\begin{proposition}{HW5.1} The integer $1$ is not divisible by $2$. That is,
$2\not\mid 1$.
\end{proposition}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{proposition}{HW5.2} Let $A=\{3x-1:x\in\Ints\}$ and
  let $B=\{3x+8:x\in\Ints\}$.  Then $A=B$.
\end{proposition}
\begin{proof}
Your proof goes here.  This proposition is admittedly not very interesting; 
I have assigned so you can practice showing that two sets are the same.
Follow the template from our proof in class of Proposition 2.13.
\end{proof}

\begin{proposition}{2.21} There are no integers $x$ such that
  $0<x<1$.
\end{proposition}

\begin{corollary}{2.22} Let $n\in\Ints$.  There are no integers $x$
  such that $n<x<n+1$.
\end{corollary}

\begin{proposition}{2.23} Let $m,n\in\Nats$.  If $n$ is divisible by $m$,
  then $m\le n$.
\end{proposition}

\begin{proposition}{2.24} For all $k\in\Nats$, $k^2+1>k$.
\end{proposition}

\begin{proposition}{2.27} For all $k\in\Ints$ such that $k\ge 2$, 
  $k^2<k^3$.
\end{proposition}

\begin{project}{2.35} Compute $\gcd(4,6)$, $\gcd(7,13)$, $\gcd(-4,10)$
  and $\gcd(-5,-15)$.  You do {\bf NOT} have to prove that you have
  found the $\gcd$.  But you do have to exhibit the integers $x$ and $y$
  in the definition of the $\gcd$.
\end{project}

\end{document}
