\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 2}
\date{Due date: Sept 10, 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 $8a+3$ and $5a+2$ are coprime for all integers $a$.   

\item Show that there are infinite many primes of the form $3k+2$. 

\item Let $S=\{\log_{10} p:  p \text{ prime}\}$. Prove that the elements of $S$ are linearly independent over $Q$, the rationals. 

\item Show that $(ac, bc)=c(a,b)$ for any integers $a, b, c$. 

\item Show that $\sqrt{2}+\sqrt{3}$ and $\log_{10} 5432$ are irrational. 

\item Show that $11 | a^2+5 b^2$ if and only if $11 | a$ and $11 | b$ for $a,b\in Z$. 


\end{enumerate}

\end{document}
