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\lhead{Mathematics and Digital Art}
\chead{}
\rhead{Number Theory Quiz}
\lfoot{Matsko}
\cfoot{}
\rfoot{29 November 2017}

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\begin{document}

\begin{enumerate}
\item Decide if the following are true or false.
\begin{enumerate}
\item $7\mid -7.$
\vskip 0.4in

\item $-5$ (mod 6) $=5.$
\vskip 0.4in
\item $-1\mid 0.$
\vskip 0.4in
\item If $a\mid b$ and $a\mid c,$ then $a\mid(bc),$ where $a,b,c$ are integers.
\vskip 0.4in
\item $8\equiv0$ (mod 3).
\vskip 0.4in
\item $13\mid91.$
\vskip 0.4in
\end{enumerate}







\item Find the following.
\begin{enumerate}
\item $53$ (mod 7).
\vskip 0.6in

\item $-97$ (mod 5).
\vskip 0.6in

\item 1,000 (mod 6).
\vskip 0.6in

\item $(-7)^{52}$ (mod 8).
\vskip 0.6in

\item $(16k-11)$ (mod 4), where $k$ is any integer.

\end{enumerate}
\eject

\item $a=75,$ and $b=3^m5^n,$ where $m,n>0.$  If $a\mid b,$ what can you conclude about $m$ and $n$?

\vskip 1.5in

\item Suppose you take an integer and raise it to the $4^{\rm th}$ power.  What are the possible digits a $4^{\rm th}$ power can end in?  Hint:  Use arithmetic (mod 10).
\vskip 3in
\item Find a number $n$ such that $n\equiv1$ (mod 6) and $n\equiv2$ (mod 5).  Then find {\em every} number $n$ with this property.
\end{enumerate}


\end{document}

