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\title{Math 412  Homework 5}
\date{Due date: Oct 1, 2010}
\author{your name}


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\maketitle

Solve the following problems. Please remember to use complete sentences
and good grammar. 



\begin{enumerate}
\item for $k\in \N$ and $\alpha\in \R$, show that $\lfloor k\alpha\rfloor\ge k\lfloor \alpha\rfloor$. 

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

\item  let $r_1, r_2, \ldots, r_{\phi(m)}$ be a reduced system of residues modulo $m=2^l$, where $l\ge 3$, then $\prod_{i} r_i\equiv 1\pmod{2^l}$.

\item Let $m\ge 3$, $r_1, r_2, \ldots, r_m$ and $r_1', r_2', \ldots, r_m'$ are two complete system of residues modulo $m$.  Show that $r_1r_1', r_2r_2', \ldots, r_mr_m'$ is a not a complete system of residues modulo $m$.  

\item Determine the order of the following $9$ modulo $25$. 

\item Let $a$ be an odd integer and integer $l\ge 3$.  Show that if the order of $a$ modulo $2^l$ is a divisor of of $2^{l-2}$.  (In other words,  $a^{2^{l-2}}\equiv 1\mod 2^l$.)










\end{enumerate}

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