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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Jan. $28^\mathrm{th}$, 2022}

%\vspace{0.2in}
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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 14.
\begin{enumerate}[(a)]
   \item  Fill out the following table (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|c|c|} \hline
element $x=$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \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 $(\mathbb{Z}_{14}, \,+ \text{ mod } 14)$ whereas the multiplicative inverses and orders go with the group structure of $(U(14), \,\cdot \text{ mod } 14)$.
   
   \item Compute \quad $3^{-5} \cdot (4-12) + 2^{123}$ \quad (mod $14$).          
             
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   \item Let $A = \begin{bmatrix} 3 & 4 \\ 3 & 1 \end{bmatrix}$. 
   
   Explain why $A$ does not belong to $\mathrm{GL}_2(\mathbb{Z}_6)$. 
   
   Then explain why $A$ does belong to $\mathrm{GL}_2(\mathbb{Z}_{14})$ and compute $A^{-1}$.
   \end{enumerate}

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\item[\#2] Use extended Euclidean Algorithm to find $15^{-1}$ in $U(1234)$.

Show your work. Simplify your answer (your answer should be a number between 0 and 1233).

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\item[\#3] Let $a,b,n \in \mathbb{Z}$ and $n>1$. Suppose that $\mathrm{gcd}(a,n)=1$ and $\mathrm{gcd}(b,n)=1$. Show that $\mathrm{gcd}(ab,n)=1$.

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\item[\#4] Let $a,b,x,y \in \mathbb{Z}$. Suppose that $ax+by=10$. 

What can be said about $\mathrm{gcd}(a,b)$? 

If $\mathrm{gcd}(x,y)=10$, what can be said about $\mathrm{gcd}(a,b)$?

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

\fbox{\parbox{6in}{
    Let $G$ be a group with identity $e \in G$. Suppose that $g^2=e$ for all $g \in G$. 
   \begin{enumerate}[(a)]
   \item What can be said about inverses of elements in $G$? What can be said about orders of elements?
   \item Prove that $G$ must be abelian.
   \end{enumerate}
}}

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


\end{description}



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