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\title{Math 214 -- Foundations of Mathematics\\
Homework 7\\
{\large{\bf Due Oct 26, 2012}}}
\date{}
\author{Your name}


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Solve the following problems.  Show all your work.   Four points each if not specified. 

\begin{enumerate}
\item Find the last digit of $7^{2011}$. 

\item Without using induction, prove that $(10^{n+1}-9n-10)\equiv 0 \pmod{81}$ for every nonnegative integer $n$.  (Remark: this is the same problem, in different form, in a previous homework set. But here you are not allowed to use induction.)

\item (6 points) Solve the following problems in $\Z_n$. 
\begin{enumerate}
\item In $\Z_8$, express the following sums and products as $[r]$, where $0\le r<8$:  
 $$[3]+[6],   [3][6],  [-13]+[138], [-13][138]$$
\item Let $[a], [b]\in \Z_8$. If $[a][b]=[0]$, does it follow that $[a]=[0]$ or $[b]=[0]$?
\item Prove that for any prime $p$,  if $[a], [b]\in \Z_p$, then $[a][b]=[0]$ implies $[a]=[0]$ or $[b]=[0]$.\\
\end{enumerate} 



\item Prove that the multiplication in $\Z_n$, $n\ge 2$, defined by $[a][b]=[ab]$ is well-defined. 

\item A relation $R$ is defined on $\Z$ by $(a, b)\in R$ if $|a-b|\le 2$.  Which of the properties reflexive, symmetric, and transitive does the relation $R$ possess? Justify your answers.\\

\item Let $R$ be a relation defined on $Z-\{0\}$ by $(a,b)\in R$ if $ab>0$.  Show that $R$ is an equivalence relation on $Z-\{0\}$. \\


\end{enumerate}

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