This is TeX, Version 3.14159 (Web2C 7.3.1) (format=tex 2000.7.19)  17 FEB 2002 00:03
**\nonstopmode\input solutions.tex
(solutions.tex
! Undefined control sequence.
l.1 \chapter
            {Solutions to Exercises}
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.7 \begin
          {solutions}{ch:primes}{1}
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.8 \item[\chap
               .\ref{ex:binomdiv}]
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.8 \item[\chap.\ref
                    {ex:binomdiv}]
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<recently read> \binom 
                       
l.10 $\binom
            {n}{k}$, for $0\leq k \leq n$, are integers.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<recently read> \binom 
                       
l.11 For the base case, note that $\binom
                                         {1}{0}=\binom{1}{1}=1$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.11 ... base case, note that $\binom{1}{0}=\binom
                                                  {1}{1}=1$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<recently read> \binom 
                       
l.12 Next, suppose that $\binom
                               {n}{k}$ is an integer for $0\leq k\leq n$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<recently read> \binom 
                       
l.13 Then $\binom
                 {n+1}{0}=\binom{n+1}{n+1}=1$, and for $0\leq i\leq n-1$,
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.13 Then $\binom{n+1}{0}=\binom
                                {n+1}{n+1}=1$, and for $0\leq i\leq n-1$,
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.14 \begin
           {align*}
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.15   \binom
             {n}{i} + \binom{n}{i+1} &=
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.15   \binom{n}{i} + \binom
                            {n}{i+1} &=
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Misplaced alignment tab character &.
l.15   \binom{n}{i} + \binom{n}{i+1} &
                                      =
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

! Undefined control sequence.
l.16    \frac
             {n!}{i!(n-i-1)!}
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \left 
l.17    \left
             (\frac{1}{n-i} + \frac{1}{i+1}\right)\\
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.17    \left(\frac
                   {1}{n-i} + \frac{1}{i+1}\right)\\
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.17    \left(\frac{1}{n-i} + \frac
                                   {1}{i+1}\right)\\
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Runaway argument?
 &= \frac {(n+1)!}{(i+1)!}{(n-i)!} = \binom {n+1}{i+1}, \end {align*}\ETC.
! Paragraph ended before \\ was complete.
<to be read again> 
                   \par 
l.21 
     
I suspect you've forgotten a `}', causing me to apply this
control sequence to too much text. How can we recover?
My plan is to forget the whole thing and hope for the best.

! Missing $ inserted.
<inserted text> 
                $
<to be read again> 
                   \par 
l.21 
     
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.23 ...ominator.  By assumption, $r<p$ so $p\nmid
                                                   r!$, and similarly
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.28 \item[\chap
                .\ref{ex:gcds}]
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.28 \item[\chap.\ref
                     {ex:gcds}]
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.31   \gcd(15,35) =\gcd(15,5)=\gcd(5,0)=\mathbf
                                                {5}.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.34 \item[\chap
                .\ref{ex:sumn}]
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.34 \item[\chap.\ref
                     {ex:sumn}]
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.35 \begin
           {enumerate}
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.38   1 + \cdots + (n+1) = \frac
                                 {n(n+1)}{2}+(n+1) = \frac{(n+1)(n+2)}{2},
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.38 ...s + (n+1) = \frac{n(n+1)}{2}+(n+1) = \frac
                                                  {(n+1)(n+2)}{2},
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.41 ...erms as $(1-2)+(3-4)+\cdots+(n-1-n)=-\frac
                                                  {n}{2}$,
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.43 $1+(-2+3)+(-4+5)+\cdots+(-(n-1)+n)=\frac
                                             {n+1}{2}$, so
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.45 $(-1)^{n+1}\lceil\frac
                           {n}{2}\rceil$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

[1] )
Output written on solutions.dvi (1 page, 1968 bytes).