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\documentclass{math215}

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\doclabel{Math 215: Homework 10}
\docauthor{Your name here}
\docdate{April 6, 2012}

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\begin{document}

\begin{project}{6.27} Finish your work on this project.
Specifically, state and prove a proposition that
characterizes the set of $n\ge 2$ such that
$\Ints_n$ satisfies the cancellation property (Axiom 1.5).  You can
opt to prove either Axiom 1.5 or its equivalent version Proposition 1.26.
\end{project}

\begin{lemma}{6.A} Suppose $n\in \Nats$, $a,b\in\Ints$, 
$2\le a\le n-1$, and $ab=n$.  Then
\[
2\le b\le n-1.
\]
\end{lemma}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{proposition}{6.28} Every integer greater than or equal to 2 can be
factored in to primes.
\end{proposition}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{lemma}{6.B}  Suppose $a,b\in \Ints$, $g\in \Ints_{\ge 0}$,
and
\begin{itemize}
  \item[(1)] $g\mid a$ and $g\mid b$
  \item[(2)] For all $d\in\Ints$ such that $d\mid a$ and $d\mid b$, $d\mid g$.
\end{itemize}
Then $g=\gcd(a,b)$.
\end{lemma}
\begin{proof}
Your proof goes here.
\end{proof}
  
\begin{lemma}{6.C}  Suppose $p$ is prime and $a\in\Ints$.  
  Then either $p\mid a$ or $\gcd(p,a)=1$.
\end{lemma}
\begin{proof}
Your proof goes here.
\end{proof}

\end{document}