\documentclass[10pt]{article} % 
\usepackage{fullpage}
\usepackage{graphicx}
\usepackage{graphics}
\usepackage{psfrag}
\usepackage{amsmath,amssymb}
\usepackage{enumerate}

\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9in}

\newcommand{\cP}{\mathcal{P}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}

\title{Math 412  Homework 4}
\date{Due date: Sept 24, 2010}
\author{your name}


\begin{document} 
\maketitle

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



\begin{enumerate}
\item (4 points) Prove that $\phi(mn)=(m,n)\phi([m,n])$. 

\item (4 points) Find all positive integers $n$ so that $\phi(n)|n$.  

\item (4 points) Let $(m,n)=1$.  Show that $m^{\phi(n)}+n^{\phi(m)}\equiv 1\pmod{mn}$. 

\item (4 points) Find the last two digits of $a_{1001}$ if $a_1=7, a_n=7^{a_{n-1}}$. 



\end{enumerate}

\end{document}
