\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
%if you put % in front of something it is a comment and doesn't affect the output

% The next 5 commands just make the margins how I personally like them. You can play with the numbers to see how it changes the text on the page
\oddsidemargin 0.0truein
\evensidemargin 0.0truein
\textwidth 6.3truein
\textheight 9truein
\topmargin -0.5in

\setlength{\parindent}{0pt} % this just makes is so the paragraphs don't indent - if you want it to indent, delete this line




\begin{document}  % you  must always start with \begin{document}

\title{MTH 346: Example Lab}
\maketitle


% adding blank lines here in your code does not produce blank lines in the output. You must use something like the \vspace command below. 

\vspace{.2in} % \vspace stands for vertical space, the length of the space is the argument in curly brackets



\section{Introduction}

Don't forget to put in an introduction. The introduction should be self contained and make sense to someone who has not done the lab. It should briefly summarize what your lab report is about. Do not use abbreviations before you define them. \\ % The \\ ends the line and signals to latex that you want to start a new line

In this sample lab I am going to write a few proofs of some problems so that you see how to use some basic latex commands. You will notice as you look at the code that math equations always go between two dollar signs.\\

\section{Research Question 1 (RQ1)} You can just type the research question here and answer it below. The answers to your research questions will be in the form of conjectures and Theorems. Instead of using ``Research Question 1" as my title you should make it more descriptive of what you are doing.\\

{\bf Research Question 1} The first research question (in my made up lab) asked me to figure out if there are a finite or infinite number of prime numbers. Before I answer that I need some notation and definitions. I will make a subsection of this section to help organize things more clearly.

\subsection{Notation and Definitions}
% note I am putting the $ around my math variables because then it will use "math mode" and the typeset will be a bit different than the other latex signalling that they are math variables
{\bf Definition 1:}
We say $a$ divides $b$ and write $a \mid b$  if there exists an integer $c$ such that $b = ac$. For example $2 \mid 6$ since $6 = 2 \cdot 3$ and $ 3 \in \mathbf{Z}$. One way to remember how to write this is that the smaller number goes first. For example, I would not write $6 \mid 2$.\\

{\bf Definition 2:} A prime number is a whole number greater than $1$ which is divisible by only itself and $1$.\\

{\bf The Fundamental Theorem of Arithmetic:} Each integer $n > 1$ can be expressed as powers of prime numbers in the form

$$n=p_1^{e_1}p_2^{e_2}\cdots p_n^{e_n}$$  % when I enclose an equation between two sets of dollar signs latex centers it on its own line

where $p_i$ are distinct prime numbers and $e_i$ are positive integers for $i=1..n$. Further, other than the order of the prime numbers, this factorization is unique.


{\subsection{Examples} 
Often when you are trying to figure out the answer to your research question you do some examples that are important to lead you to a conjecture. \\

In my case, I notice that it seems like there are a lot of primes. I can think of a  lot of them: $3,5,7,11,...$ it seems to go on forever. I think there are an infinite number of primes. I will prove it next.


\subsection{Answer}

{\bf Theorem:} There are an infinite number of prime numbers.\\


{\bf Proof:} I will prove this by contradiction (This is Euclid's proof).\\
 
 Assume there are only a finite number of primes, say $n$ prime numbers. Let us list the primes as $p_1, p_2, p_3,...,p_n.$ \\

Now consider the number consisting of all of the primes multiplied together plus one: $$Q = p_1p_2\cdots p_n +1.$$

 Clearly $Q$ is an integer since multiplication and addition of integers is closed. Therefore the Fundamental Theorem of Arithmetic applies and $Q$ can be factored into a product of powers of distinct prime numbers. Since $p_1,...,p_n$ are the only prime numbers there must be at least one of these that divides $Q$. Without loss of generality, let's suppose that $p_i \mid Q$. This means that $Q = p_i \cdot r$ for some $r \in \mathbf{Z}$.\\

Now Consider $Q-1 = p_1p_2\cdots p_i \cdots p_n$. Clearly $p_i$ also divides $Q-1$ so we'll write $Q-1 = p_1 \cdot s$ ($s \in \mathbf{Z}$).\\

Finally consider $Q - (Q-1) = 1$. We can substitute on the left hand side to get:\\

$\begin{array}{rl}
Q - (Q - 1) = & \\
p_i \cdot r - p_i \cdot s  = & \\
p_i(r-s)  = & 1. \\
\end{array}$

Therefore by the definition of divides, $p_i \mid 1$. However the only whole number dividing $1$ is $1$ implying $p_i = 1$ but we assumed $p_i$ was prime and $1$ is not prime. Thus we have a contradiction and our assumption that there are only a finite number of prime numbers is false so there must be an infinite number of prime numbers. $\Box$

\section{Exercise}

{\bf Exercise 1:} What are the prime divisors of $50$? Find the unique prime power factorization of $50$. \\

{\bf Answer:} The prime divisors of $50$ are $\{2,5\}$. The unique prime power factorization is $50=2\cdot5^2$.\\

\section{Conclusion}

Put some sort of conclusion here.




\end{document} % you must always end with \end{document}
