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\title{\sc Math 2100 / 2350 -- Homework 4}
\author{
	Fall 2020\\[10pt]
	due Wednesday, \textbf{October 28},
	\underline{on D2L, by the beginning of class}\\[10pt]
	Sections 2.1, 2.2, 2.3, 2.4
}
\date{}
\begin{document}
\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 on D2L at the \emph{beginning} of class. It must be typed in Latex (other formats such as Word are not acceptable). \textbf{You must submit the .pdf file, but you do not have to submit the .tex file unless I ask for it} Any pictures can be drawn by hand and added to the Latex file with the ``$\backslash$includegraphics'' command (see how I do it in this document). Please write the questions in the correct order. Explain all reasoning.

\textbf{Note:} You can write a proof in Latex with the command
\begin{verbatim}
\begin{proof}
    proof goes here	
\end{proof}
\end{verbatim}

\begin{enumerate}	
	\item Decide if the following statement is true or false. If it's true, prove it. If it's false, provide a counterexample.
	\begin{center}
		If $A$, $B$, and $C$ are integers and if $A$ divides $B$ and $B$ divides $C$, then $A$ divides $C$.		
	\end{center}

	\item Decide if the following statement is true or false. If it's true, prove it. If it's false, provide a counterexample.
	\begin{center}
		If $A$, $B$, and $C$ are integers and if $A$ divides $B$ and $A$ divides $C$, then $A\cdot B$ divides $C$.		
	\end{center}

	\item Decide if the following statement is true or false. If it's true, prove it. If it's false, provide a counterexample.
	\begin{center}
		If $n$ is a positive even integer, then $3^n+1$ is divisible by $5$.
	\end{center}
	
	\item Decide if the following statement is true or false. If it's true, prove it. If it's false, provide a counterexample.
	\begin{center}
		If $n$ is a positive even integer, then $n^3 + 2n$ is divisible by $4$.
	\end{center}
	
	\item Decide if the following statement is true or false. If it's true, prove it. If it's false, provide a counterexample.
	\begin{center}
		If $m$ is a positive odd integer, then $m^2 - 1$ is divisible by $8$.
	\end{center}
	
	\item Prove that if $3$ divides $4^{n-1}-1$ then $3$ divides $4^n-1$.

	\item Prove that for all positive integers $n$,
	\[
		\ds\sum_{k=0}^n (k \cdot k!) = (n+1)! - 1.
	\]
		
	\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 $n \geq 4$,
	\[
		n! > 2^n.
	\]

\end{enumerate}
\end{document}


