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\title{Math 412  Homework 3}
\date{Due date: Sept 18, 2015}
\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 $\frac{(2n)!}{(n!)^2}$ is even.  

\item Show that $x^2-2y^2=77$ has no integer solution.  

\item Solve the following system of linear congruences: $5x\equiv 1\pmod{9}, x\equiv 8\pmod{15}, x\equiv 3\pmod{25}$. 

\item Show that the system of congruences $x\equiv a_1\pmod{m_1},  x\equiv a_2\pmod{m_2}$ has a solution if and only if $(m_1, m_2)| (a_1-a_2)$.  Show that when there is a solution, it is unique modulo $[m_1, m_2]$. 


\item Solve the equation $x^2+5x+13\equiv 0\pmod{3^4}$. 

\item Show that $n, n+2$ are both prime if and only if $$4((n-1)!+1)\equiv -n\pmod{n(n+2)}.$$



\end{enumerate}

\end{document}
