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\title{Math 412  Homework 1}
\date{Due date: Sept 3, 2010}
\author{your name}


\begin{document} 
\maketitle

Solve the following problems. Please remember to use complete sentences
and good grammar. Each problem is 4 points.



\begin{enumerate}
\item Show that if $2 | n$ and $3 | n$, then $6 | n$.  Prove that $6 | (n^3-n)$ for all positive integer $n$. 

\item Show that the square of every odd integer is of the form $8k+1$.  

\item Show that if $p_k$ is the $k$-th prime, where $k$ is a positive integer, then $p_n\le p_1p_2\ldots p_{n-1}+1$ for all integers $n$ with $n\ge 3$. 

\item Show that the integer $Q_n=n!+1$, where $n$ is a positive integer, has a prime divisor greater than $n$.  Conclude that there are infinitely many primes.  

\item Find integers $m$ and $n$ so that $(2106, 8318)=2106m+8318n$. 

  
\end{enumerate}

\end{document}
