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\noindent
\parbox{1.95in}{\bf Math 3110}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{1.95in}{\bf \hfill Due: Fri., Sept. 6${}^\mathbf{th}$\!, 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 whatever.
\begin{enumerate}[(a)]
   \item  Working mod 12. Create (and fill) a table as follows\\ \force\hfill (if an entry is undefined, write ``DNE'' = does not exist):

\begin{tabular}{|r||c|c|c|c|c|c|c|c|c|c|c|c|} \hline
element $x=$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \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 $(\mathbb{Z}_{12}, \,+ \text{ mod } 12)$ whereas the multiplicative inverses and orders go with the group structure of $(U(12), \,\cdot \text{ mod } 12)$.
   
   \item Compute \quad $2^{100} + (13-26) \cdot 5^{-2}$ \quad (mod $8$).          
             
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   \item Let $A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$. 
   
   Explain why $A$ does not belong to $\mathrm{GL}_2(\mathbb{Z}_{12})$. 
   
   Then explain why $A$ does belong to $\mathrm{GL}_2(\mathbb{Z}_{7})$ and compute $A^{-1}$.
   \end{enumerate}

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\item[\#2] Use the extended Euclidean Algorithm to find $109^{-1}$ in $U(4115)$.

Show your work. Simplify your answer (your answer should be a number between 0 and 4114).

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\item[\#3] Let $a,b \in \mathbb{Z}$ (not both zero) such that $d=\mathrm{gcd}(a,b)$. In addition, let $a=da'$ and $b=db'$.\\ \force\hfill Show that $\mathrm{gcd}(a',b')=1$.

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\item[\#4] Show that for every $n \in \mathbb{Z}$, we have $n^3=n$ (mod 6).

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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 $(ab)^{-1} \not= a^{-1}b^{-1}$.
   
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   \item Prove $G$ is an abelian group if and only if for all $a,b \in G$, $(ab)^{-1}=a^{-1}b^{-1}$.
   
   \end{enumerate}
}}

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


\end{description}



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