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\noindent\emph{\textbf{Score: \underline{\hspace*{1cm}} out of 10\hfill Name: \underline{\hspace*{4.1cm}}}}

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\bf\emph{MATH 221 \--- Foundations of Mathematics \--- Dr. Russell E. Goodman\\
Fall 2010 \--- Take-Home Quiz \#8}\\
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\noindent\emph{This quiz is worth 10 points. Consider the theorem stated below along with the 12 sentences provided on the second page.  Rearrange the sentences so the resulting text forms a logical and well-organized proof of the theorem.  Please \textbf{literally} cut out the 12 sentences and tape them to this first page in the order you feel is appropriate and turn in the taped quiz \textbf{by 4pm on December 8}.}

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\noindent\emph{\textbf{You are expected to work alone on this quiz, neither giving nor receiving assistance to/from any classmates, friends or professors. Any violation of this expectation will result in a grade of zero on this quiz for all involved parties.}}

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\noindent\textbf{Theorem:} \emph{Every integer greater than 1 is expressible as a product of primes.}

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\begin{enumerate}
\item Hence, we see that $n=\left( (p_{1}\cdot p_{2}\cdots p_{s})\cdot(q_{1}\cdot q_{2}\cdots q_{t})\right)$.
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\item But $a$ and $b$ must then be expressible as products of primes due to the minimality of $n$.
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\item It also must be the case that $1<b<n$.
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\item Then $n$ must be a composite number (non-prime).
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\item Suppose there are some positive integers greater than 1 that are not expressible as products of primes.
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\item Thus, $a=p_{1}\cdot p_{2}\cdots p_{s}$ and $b=q_{1}\cdot q_{2}\cdots q_{t}$ where all $p_{i}$ and $q_{j}$ are primes and $s,t\ge 1$.
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\item By the Well-Ordering Principle, there must be a smallest such integer $n$.
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\item Therefore, it must be the case that every integer greater than 1 is expressible as a product of primes.
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\item As a result, there must be an integer $a$ such that $1<a<n$ and such that $a|n$.
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\item We proceed using proof by contradiction.
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\item Using the definition of divisibility, there is an integer $b$ with $a\cdot b=n$.
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\item This is contrary to our assumption about $n$.
\end{enumerate}
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