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\title{\sc Math 2100 / 2350 -- Homework 4}
\author{
	Fall 2019\\[10pt]
	due Wednesday, \textbf{October 30}, at the beginning of class\\[10pt]
	Sections 2.2, 2.3, 2.4, half of 2.5
}
\date{}
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\maketitle

\begin{center} 
	\emph{This homework assignment was written in \LaTeX{}. You can find the source code on the course website.}
\end{center}

\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. \textbf{Staple your work} together (do not just fold over the corner). Please write the questions in the correct order. If I cannot read your handwriting, you won't receive credit. Explain all reasoning.

\begin{enumerate}
	\item Decide if the following statement is true. If it is, prove it. If it's not, provide a counterexample.
	\begin{center}
		For integers $x$, $y$, and $z$, if $x$ divides $y$ and $x$ divides $z$, then $x^2$ divides $yz$.
	\end{center}
	
	\item Decide if the following statement is true. If it is, prove it. If it's not, provide a counterexample.
	\begin{center}
		For integers $x$, $y$, and $z$, if $x$ divides $z$ and $y$ divides $z$, then $xy$ divides $z^2$.
	\end{center}
	
	\item Prove that for all positive integers $n$,
	\[
		1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2.
	\]
	
	\item Prove that for all positive integers $n$,
	\[
		\sum_{k=0}^n (k \cdot k!) = (n+1)! - 1.
	\]
	
	\item Prove that for all positive integers $n \geq 2$, the number $2^{3n}-1$ is not prime.
	
	\item Prove that for all positive integers $n \geq 4$,
	\[
		n! > 2^n.
	\]
	
	\item Prove that at a completely full Milwaukee Bucks game at the Fiserv Forum, there \emph{must} be at least two people that have both the same birthday \emph{and} the same first initial. (Note: you will have to look up the capacity of the arena!)

	\item Use the pigeonhole principle to prove that given any five integers, there will be two that have a sum or difference divisible by 7.
		
\end{enumerate}
\end{document}


