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\headers{Math373}{Homework 4}{Due Fri, Feb. 7, 2014}

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\noindent
\textbf{Directions:} Solve the following problems.  All written work must be your own.  See the course syllabus for detailed rules.

\begin{enumerate}

\item {}[JJJ 1.28] Compute the following values of the order function.
\begin{enumerate}
	\item $\ord_2 (2816)$
	\item $\ord_7 (2222574487)$
	\item $\ord_p (46375)$ for each prime $p\in \{3,5,7,11\}$.
\end{enumerate}

\item {} [JJJ 1.29] Let $p$ be a prime number, and let $a$ and $b$ be positive integers.  Prove the following.
\begin{enumerate}
	\item $\ord_p(ab) = \ord_p(a) + \ord_p(b)$
	\item $\ord_p(a+b) \ge \min\{\ord_p(a), \ord_p(b)\}$
	\item If $\ord_p(a) \ne \ord_p(b)$, then $\ord_p(a+b) = \min\{\ord_p(a), \ord_p(b)\}$.
\end{enumerate}
\end{enumerate}
\end{document}
