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\noindent
\parbox{1.95in}{\bf Math 3110}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{1.95in}{\bf \hfill Due: Fri., Feb. $2^\mathrm{nd}$, 2024}

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%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

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\noindent
Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force

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\begin{description}
\item[\#1] Workin' mod 9.
\begin{enumerate}[(a)]
   \item  Create (and fill) a table as follows (if an entry is undefined, write ``DNE'' = does not exist):

\begin{tabular}{|r||c|c|c|c|c|c|c|c|c|} \hline
element $x=$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \Large\strut \\ \hline
additive inverse $-x=$ & 0 &  &  &  &  &  &  &  &  \Large\strut \\ \hline
additive order $|x|=$ & 1 &  &  &  &  &  &  &  &  \Large\strut \\ \hline
mutiplicative inverse $x^{-1}=$ & DNE &  &  &  &  &  &  &  & \Large\strut \\ \hline
multiplicative order $|x|=$ & DNE &  &  &  &  &  &  &  &   \Large\strut \\ \hline
\end{tabular}

{\it Note:} Additive inverses and orders go with the group structure of $(\mathbb{Z}_{9}, \,+ \text{ mod } 9)$ whereas the multiplicative inverses and orders go with the group structure of $(U(9), \,\cdot \text{ mod } 9)$.
   
   \item Compute \quad $3^{1234} + (11-33) \cdot 2^{-4}$ \quad (mod $9$).          
             
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   \item Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. 
   
   Explain why $A$ does not belong to $\mathrm{GL}_2(\mathbb{Z}_8)$. 
   
   Then explain why $A$ does belong to $\mathrm{GL}_2(\mathbb{Z}_{9})$ and compute $A^{-1}$.
   \end{enumerate}

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\item[\#2] Use the extended Euclidean Algorithm to find $773^{-1}$ in $U(2332)$.

Show your work. Simplify your answer (your answer should be a number between 0 and 2331).

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\item[\#3] Let $a,b,x,y \in \mathbb{Z}$ such that $ax+by=15$. What can be said about $\mathrm{gcd}(a,b)$? 

Suppose in addition we have $\mathrm{gcd}(x,y)=15$, then prove $a$ and $b$ are relatively prime.

   \vspace{0.2in}

% Note to self: should be b and c divide a so bc divides a. This version is silly.
\item[\#4] Let $a,b,c \in \mathbb{Z}$. Assume $a$ divides $b$ and $a$ divides $c$. Also, suppose $b$ and $c$ are relatively prime (i.e., $\mathrm{gcd}(b,c)=1$).
Show $a$ divides $bc$. 

{\it Hint:} Use an integer linear combination involving $b$ and $c$ then multiply that by $bc$. 

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\item[\#5 RESUBMIT] Type up Homework \#1 Problem \#2 and its solution in \LaTeX. 

\fbox{\parbox{6in}{Let $G$ be a group with identity $e \in G$. 
   \begin{enumerate}[(a)]
   \item Give a concrete example of a group $G$ and elements $a,b \in G$ where $(ab)^2 \not= a^2b^2$.
   
   \vspace{0.05in}
   
   \item Prove $G$ is an abelian group if and only if for all $a,b \in G$, $(ab)^2=a^2b^2$.
   
   \end{enumerate}
}}

     When typing this problem up, write it up carefully: Restate the problem. Write in complete sentences.  


\end{description}



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