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\title{\sc Math 2100 / 2105 / 2350 -- Homework 7}
\author{due Thursday, \textbf{October 25}, at the beginning of class}
\date{}
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\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 Prove that for any integer $n$, if $n^2$ is odd, then $n$ is odd.
	\item Suppose $X$, $Y$, and $Z$ are integers. Prove that if $X$ divides $Y$ and $Y$ divides $Z$, then $X$ divides $Z$.
	\item Decide if the following statement is true. If it is, prove it. If it's not, provide a counterexample.
	\begin{center}
		For all integers $m$, $3$ divides $m^3 - m$.
	\end{center}
	\item Prove that all integers are rational numbers. 
	\item Decide if the following statement is true. If it is, prove it. If it's not, provide a counterexample.
	\begin{center}
		If $n$ is a positive even integers, then $3^n+1$ is divisible by $5$.
	\end{center}
	\item Decide if the following statement is true. If it is, prove it. If it's not, provide a counterexample.
	\begin{center}
		If $n$ is a positive even integer and $n \geq 4$ then $2^n-1$ is not prime.
	\end{center}
	
\end{enumerate}
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