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\centerline{\bf Homework 1 Morally Due Feb 6}

\begin{enumerate}
\item
(0 points)
If you are not getting emails that the class gets, then email Bill as soon as possible. 

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\item
(30 points) List all of the elements of $\{0,1,\ldots,20\}$ that have multiplicative inverses mod 21. 
For each such element also give the inverse.

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\item
(40 points)
The alphabet is $\{0,\ldots,9\}$.
We interpret the input as a base 10 natural number, read \emph{right to left}.
So the number 29139 will be read 9-3-1-9-2.

\begin{enumerate}
\item
(10 points) Compute 

$10^0 \pmod {14}$

$10^1 \pmod {14}$

$10^2 \pmod {14}$

etc. 

\noindent
until you spot a pattern. What is the pattern?

\item
(10 points) Recall that 

$$a_k\times 10^k +  a_{k-1}\times 10^{k-1} +  \cdots + a_0\times 10^0 \equiv a_k + \cdots + a_0 \pmod 9.$$

Come up with a statement of that type for mod 14. 
You may use DOT-DOT-DOT and you may have a set of cases. 

\item
(20 points) Let 

$$
A= \{ x \colon x\equiv 5 \pmod {14} \} \cup \{ x \colon x\equiv 7 \pmod {14} \}.
$$

($x$ is in base 10.)

Give the DFA for $A$ by giving the transition table and
specifying what are the start state is and what the final states are. 

{\bf ADVICE} For the transtion table {\bf do not} have LOTS of rows. You can have things like
(this is not the actual answer)

For all $\sigma\in \{0,\ldots,9)$, $\delta(q,\sigma)=x+8\sigma \pmod {87}$. 

There may be cases. 


\end{enumerate} 


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% Problem 5
\item 
(30 points) We define $\#_a(w)$ to be the number of $a$'s in $w$. The alphabet is $\{a, b\}$.

\begin{enumerate}
\item
(15 points) Draw a DFA for $\{w \mid \#_a(w) \equiv 0,1 \pmod{4}\}$. How many states does this DFA have?
\item
(15 points) Draw a DFA for $\{w \mid \#_a(w) \equiv 0,2 \pmod{4}\}$. How many states does this DFA have? (Hint: this should be less states than the prior part.)
\end{enumerate}


\end{enumerate}
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