Template for MTH 210
Author
Ted Sundstrom
Last Updated
10年前
License
Other (as stated in the work)
Abstract
This template should be used for writing your solutions of the portfolio problems for MTH 210 - Communicating in Mathematics.
\documentclass[12pt]{article}
\pagestyle{myheadings}
%Enter your name, the portfolio problem number, and the draft number.
\title{Portfolio Problem 1 -- Draft 1}
\author{Ted Sundstrom}
%Enter your name, the portfolio problem number, and the draft number. This will be a heading on pages after the first page.
\markright{Ted Sundstrom Problem 1 -- Draft 1}
\usepackage{amsmath,amssymb,amsthm,amsfonts,graphics}
%The following commands allow us to typeset theorems, propositions, definitions, etc.
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem*{definition}{Definition}
\renewcommand{\qedsymbol}{\ensuremath{\blacksquare}}
\begin{document}
\maketitle
%Enter your email address.
\begin{center}
\textbf{email address: sundstrt@gvsu.edu}
\end{center}
\begin{proposition}
If $a$ and $b$ are type 2 integers, then $a \cdot b$ is a type 1 integer.
\end{proposition}
\begin{proof}
We assume that $a$ and $b$ are type 2 integers and will prove that $a \cdot b$ is a type 1 integer. Since $a$ and $b$ are type 2 integers, there exist integers $m$ and $n$ such that
\[
a = 3m + 2 \text{ ~~~~~~and~~~~~ } b = 3n + 2.
\]
We can now use substitution and algebra .........
\begin{align*}
ab &= (3m + 2)(3n + 2) \\
&= 9mn + 6m + 6n + 4 \\
&= 9mn + 6m + 6n + 3 + 1
\end{align*}
\end{proof}
\end{document}