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1  %$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_mtn0/c_mtn0.tex,v 1.4 2002/12/01 21:29:08 dtashley Exp $  %$Header$
2    
3  \chapter[\cmtnzeroshorttitle{}]{\cmtnzerolongtitle{}}  \chapter[\cmtnzeroshorttitle{}]{\cmtnzerolongtitle{}}
4    
5  \label{cmtn0}  \label{cmtn0}
6    
7  \beginchapterquote{``If intellectual curiosity, professional pride, and ambition are  \beginchapterquote{``If intellectual curiosity, professional pride, and ambition are
8                       the dominant incentives to research, then assuredly no one has                       the dominant incentives to research, then assuredly no one has
9                       a fairer chance of gratifying them than a mathematician.  His                       a fairer chance of gratifying them than a mathematician.  His
10                       subject is the most curious of all---there is none in which                       subject is the most curious of all---there is none in which
11                       truth plays such odd pranks.  It has the most elaborate                       truth plays such odd pranks.  It has the most elaborate
12                       and the most fascinating technique, and gives unrivaled                       and the most fascinating technique, and gives unrivaled
13                       openings for the display of sheer professional skill.  Finally,                       openings for the display of sheer professional skill.  Finally,
14                       as history proves abundantly, mathematical achievement, whatever                       as history proves abundantly, mathematical achievement, whatever
15                       its intrinsic worth, is the most enduring                       its intrinsic worth, is the most enduring
16                       of all.''}                       of all.''}
17                       {G.H. Hardy \cite{bibref:b:mathematiciansapology:1940}}                       {G.H. Hardy \cite{bibref:b:mathematiciansapology:1940}}
18    
19  \section{Introduction}  \section{Introduction}
20  %Section Tag: INT0  %Section Tag: INT0
21  \label{cmtn0:sint0}  \label{cmtn0:sint0}
22    
23    
24  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
25  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
26  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
27    
28  \index{floor function@floor function ($\lfloor\cdot\rfloor$)}%  \index{floor function@floor function ($\lfloor\cdot\rfloor$)}%
29  \index{--@$\lfloor\cdot\rfloor$ (\emph{floor($\cdot$)} function)}%  \index{--@$\lfloor\cdot\rfloor$ (\emph{floor($\cdot$)} function)}%
30  \index{ceiling function@ceiling function ($\lceil\cdot\rceil$)}%  \index{ceiling function@ceiling function ($\lceil\cdot\rceil$)}%
31  \index{--@$\lceil\cdot\rceil$ (\emph{ceiling($\cdot$)} function)}%  \index{--@$\lceil\cdot\rceil$ (\emph{ceiling($\cdot$)} function)}%
32  \section{The Floor \mbox{\boldmath $\lfloor\cdot\rfloor$} And Ceiling \mbox{\boldmath $\lceil\cdot\rceil$} Functions}  \section{The Floor \mbox{\boldmath $\lfloor\cdot\rfloor$} And Ceiling \mbox{\boldmath $\lceil\cdot\rceil$} Functions}
33  \label{cmtn0:sfcf0}  \label{cmtn0:sfcf0}
34    
35  The \emph{floor} function, denoted $\lfloor\cdot\rfloor$, is defined to return  The \emph{floor} function, denoted $\lfloor\cdot\rfloor$, is defined to return
36  the largest integer not larger than the argument.  For example,  the largest integer not larger than the argument.  For example,
37  $\lfloor 3 \rfloor = \lfloor 3.9999 \rfloor = 3$.  For negative arguments, the definition  $\lfloor 3 \rfloor = \lfloor 3.9999 \rfloor = 3$.  For negative arguments, the definition
38  is identical:  $\lfloor -4 \rfloor = \lfloor -3.9 \rfloor = -4$.  is identical:  $\lfloor -4 \rfloor = \lfloor -3.9 \rfloor = -4$.
39    
40  The \emph{ceiling} function, denoted $\lceil\cdot\rceil$, is defined to return  The \emph{ceiling} function, denoted $\lceil\cdot\rceil$, is defined to return
41  the smallest integer not less than the argument.  For example,  the smallest integer not less than the argument.  For example,
42  $\lceil 3.0001 \rceil = \lceil 4 \rceil = 4$.  For negative arguments, the definition  $\lceil 3.0001 \rceil = \lceil 4 \rceil = 4$.  For negative arguments, the definition
43  is identical:  $\lceil -4 \rceil = \lceil -4.9 \rceil = -4$.  is identical:  $\lceil -4 \rceil = \lceil -4.9 \rceil = -4$.
44    
45  Note that the definitions presented above for negative arguments  Note that the definitions presented above for negative arguments
46  differ from what is commonly implemented in spreadsheet software and other consumer  differ from what is commonly implemented in spreadsheet software and other consumer
47  software.  software.
48    
49  It can be verfied easily that for  It can be verfied easily that for
50  $a \in \vworkintsetnonneg$, $b \in \vworkintsetpos$,  $a \in \vworkintsetnonneg$, $b \in \vworkintsetpos$,
51    
52  \begin{equation}  \begin{equation}
53  \label{eq:cmtn0:sfcf0:01}  \label{eq:cmtn0:sfcf0:01}
54  \frac{a}{b} = \left\lfloor\frac{a}{b}\right\rfloor + \frac{a \bmod b}{b}  \frac{a}{b} = \left\lfloor\frac{a}{b}\right\rfloor + \frac{a \bmod b}{b}
55  \end{equation}  \end{equation}
56    
57  \noindent{}and consequently that  \noindent{}and consequently that
58    
59  \begin{equation}  \begin{equation}
60  \label{eq:cmtn0:sfcf0:02}  \label{eq:cmtn0:sfcf0:02}
61  \left\lfloor\frac{a}{b}\right\rfloor = \frac{a}{b} - \frac{a \bmod b}{b} .  \left\lfloor\frac{a}{b}\right\rfloor = \frac{a}{b} - \frac{a \bmod b}{b} .
62  \end{equation}  \end{equation}
63    
64  \noindent{}(\ref{eq:cmtn0:sfcf0:02}) is a very useful identity for  \noindent{}(\ref{eq:cmtn0:sfcf0:02}) is a very useful identity for
65  decomposing expressions involving the \emph{floor($\cdot$)} function.  decomposing expressions involving the \emph{floor($\cdot$)} function.
66    
67  \section{Tests For Divisibility Of Integers}  \section{Tests For Divisibility Of Integers}
68  %Section Tag: TDI0  %Section Tag: TDI0
69    
70  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
71  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
72  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
73  \subsection{Tests For Divisibility By 2, 3, 5, 6, 7, And 11}  \subsection{Tests For Divisibility By 2, 3, 5, 6, 7, And 11}
74    
75  It is often useful to be able to inspect a radix-10 integer and quickly  It is often useful to be able to inspect a radix-10 integer and quickly
76  determine if it can be divided by a small prime number.  This section  determine if it can be divided by a small prime number.  This section
77  presents tests which can be used to easily determine divisibility by  presents tests which can be used to easily determine divisibility by
78  2, 3, 5, 7, and 11.  2, 3, 5, 7, and 11.
79    
80  Placeholder\index{divisibility tests for integers!by 0002@by 2}  Placeholder\index{divisibility tests for integers!by 0002@by 2}
81  reserved for divisibility by 2.  reserved for divisibility by 2.
82    
83  Placeholder\index{divisibility tests for integers!by 0003@by 3}  Placeholder\index{divisibility tests for integers!by 0003@by 3}
84  reserved for divisibility by 3.  reserved for divisibility by 3.
85    
86  Placeholder\index{divisibility tests for integers!by 0005@by 5}  Placeholder\index{divisibility tests for integers!by 0005@by 5}
87  reserved for divisibility by 5.  reserved for divisibility by 5.
88    
89  Placeholder\index{divisibility tests for integers!by 0007@by 7}  Placeholder\index{divisibility tests for integers!by 0007@by 7}
90  reserved for divisibility by 7.  reserved for divisibility by 7.
91    
92  Placeholder\index{divisibility tests for integers!by 0011@by 11}  Placeholder\index{divisibility tests for integers!by 0011@by 11}
93  reserved for divisibility by 11.  reserved for divisibility by 11.
94    
95    
96  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
97  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
98  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
99  \subsection{Tests For Divisibility By  2$^N$, 6, 9, And 10$^N$}  \subsection{Tests For Divisibility By  2$^N$, 6, 9, And 10$^N$}
100    
101  Placeholder\index{divisibility tests for integers!by 0002N@by 2$^N$}  Placeholder\index{divisibility tests for integers!by 0002N@by 2$^N$}
102  reserved for divisibility by 2$^N$.  reserved for divisibility by 2$^N$.
103    
104  Placeholder\index{divisibility tests for integers!by 0006@by 6}  Placeholder\index{divisibility tests for integers!by 0006@by 6}
105  reserved for divisibility by 6.  reserved for divisibility by 6.
106    
107  Placeholder\index{divisibility tests for integers!by 0009@by 9}  Placeholder\index{divisibility tests for integers!by 0009@by 9}
108  reserved for divisibility by 9.  reserved for divisibility by 9.
109    
110  Placeholder\index{divisibility tests for integers!by 0010N@by 10$^N$}  Placeholder\index{divisibility tests for integers!by 0010N@by 10$^N$}
111  reserved for divisibility by 10$^N$.  reserved for divisibility by 10$^N$.
112    
113    
114  \subsection{David G. Radcliffe's Proof:  Rearrangement Of Digits Of $2^N$}  \subsection{David G. Radcliffe's Proof:  Rearrangement Of Digits Of $2^N$}
115  %Subsection Tag: DGR0  %Subsection Tag: DGR0
116    
117  In 07/00, Paul Harvey (\texttt{pharvey@derwent.co.uk}) made the following  In 07/00, Paul Harvey (\texttt{pharvey@derwent.co.uk}) made the following
118  post to \texttt{sci.math} \cite{bibref:n:scimathnewsgroup}:  post to \texttt{sci.math} \cite{bibref:n:scimathnewsgroup}:
119    
120  \begin{quote}  \begin{quote}
121  {I've got a little problem which is bugging me, perhaps someone out there  {I've got a little problem which is bugging me, perhaps someone out there
122  can point me in the right direction \ldots{}}  can point me in the right direction \ldots{}}
123    
124  {Does there exist a positive integer which is a power of 2, whose digits can  {Does there exist a positive integer which is a power of 2, whose digits can
125  be rearranged to give a different power of 2?}  be rearranged to give a different power of 2?}
126  \end{quote}  \end{quote}
127    
128  David G. Radcliffe \cite{bibref:i:davidgradcliffe}  David G. Radcliffe \cite{bibref:i:davidgradcliffe}
129  responded with a beautiful proof, which is presented below  responded with a beautiful proof, which is presented below
130  as a theorem.  as a theorem.
131    
132  \begin{vworktheoremstatement}  \begin{vworktheoremstatement}
133  No radix-10 positive integral power of 2 (i.e. 1, 2, 4, 8, 16, 32, etc.), with  No radix-10 positive integral power of 2 (i.e. 1, 2, 4, 8, 16, 32, etc.), with
134  any leading 0's removed, can be used  any leading 0's removed, can be used
135  to form another radix-10 positive integral power of 2 by simple rearrangement  to form another radix-10 positive integral power of 2 by simple rearrangement
136  of the digits.  of the digits.
137  \end{vworktheoremstatement}  \end{vworktheoremstatement}
138  \begin{vworktheoremproof}  \begin{vworktheoremproof}
139  Suppose that $x$ and $y$ are two different powers of 2, $y>x$, and that  Suppose that $x$ and $y$ are two different powers of 2, $y>x$, and that
140  the digits of $x$ can be rearranged to form $y$.  $y<10x$, since both  the digits of $x$ can be rearranged to form $y$.  $y<10x$, since both
141  $x$ and $y$ must have the same number of digits.  Thus, there  $x$ and $y$ must have the same number of digits.  Thus, there
142  are three possibilities, $y=2x$, $y=4x$, or $y=8x$.  are three possibilities, $y=2x$, $y=4x$, or $y=8x$.
143    
144  Since $x$ and $y$ have the same digits, but in a different order,  Since $x$ and $y$ have the same digits, but in a different order,
145  the sum of the digits of $x$ is equal to the sum of the digits of $y$.  the sum of the digits of $x$ is equal to the sum of the digits of $y$.
146  It follows that $y-x$ is divisible by 9.  (This follows because  It follows that $y-x$ is divisible by 9.  (This follows because
147  the sum of the digits of an integer $i$, summing the intermediate  the sum of the digits of an integer $i$, summing the intermediate
148  sums as many times as necessary to yield a single-digit result,  sums as many times as necessary to yield a single-digit result,
149  yield either 9 implying that $i \; mod \; 9 = 0$, or yielding $i \; mod \; 9$.  yield either 9 implying that $i \; mod \; 9 = 0$, or yielding $i \; mod \; 9$.
150  If the digits of $x$ and $y$ are the same,  If the digits of $x$ and $y$ are the same,
151  the sums of their digits are the same, thus $(x \; mod \; 9) = (y \; mod \; 9)$,  the sums of their digits are the same, thus $(x \; mod \; 9) = (y \; mod \; 9)$,
152  which implies that $((y-x) \; mod \; 9) = 0$, i.e. that $y-x$ is divisible  which implies that $((y-x) \; mod \; 9) = 0$, i.e. that $y-x$ is divisible
153  by 9.)  by 9.)
154    
155  If $y \in \{ 2x, 4x, 8x \}$, then $y-x \in \{ x, 3x, 7x \}$.  It would  If $y \in \{ 2x, 4x, 8x \}$, then $y-x \in \{ x, 3x, 7x \}$.  It would
156  follow that $x$ is divisible by 3, a contradiction.  follow that $x$ is divisible by 3, a contradiction.
157  \end{vworktheoremproof}  \end{vworktheoremproof}
158  \vworktheoremfooter{}  \vworktheoremfooter{}
159    
160  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
161  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
162  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
163  \section{The Pigeonhole Principle}  \section{The Pigeonhole Principle}
164  \label{cmtn0:sphp0}  \label{cmtn0:sphp0}
165    
166  The \index{pigeonhole principle}\emph{pigeonhole principle} is a statement  The \index{pigeonhole principle}\emph{pigeonhole principle} is a statement
167  that if $m$ items are placed into $n$ slots, with $m > n$, then at least one  that if $m$ items are placed into $n$ slots, with $m > n$, then at least one
168  slot will contain more than one item.  This is also known as  slot will contain more than one item.  This is also known as
169  \index{Dirichlet's box principle}\emph{Dirichlet's box principle}.  \index{Dirichlet's box principle}\emph{Dirichlet's box principle}.
170    
171  A related statement is that $m$ items are placed into $n$ slots,  A related statement is that $m$ items are placed into $n$ slots,
172  with $m < n$, then at least one  with $m < n$, then at least one
173  slot will be empty.  slot will be empty.
174    
175  Despite its simplicity, the pigeonhole principle is the basis for many important  Despite its simplicity, the pigeonhole principle is the basis for many important
176  proofs and observations in number theory.  proofs and observations in number theory.
177    
178    
179  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
180  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
181  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
182  \section{Exercises}  \section{Exercises}
183    
184    
185  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
186    
187  \noindent\begin{figure}[!b]  \noindent\begin{figure}[!b]
188  \noindent\rule[-0.25in]{\textwidth}{1pt}  \noindent\rule[-0.25in]{\textwidth}{1pt}
189  \begin{tiny}  \begin{tiny}
190  \begin{verbatim}  \begin{verbatim}
191  $RCSfile: c_mtn0.tex,v $  $RCSfile: c_mtn0.tex,v $
192  $Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_mtn0/c_mtn0.tex,v $  $Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_mtn0/c_mtn0.tex,v $
193  $Revision: 1.4 $  $Revision: 1.4 $
194  $Author: dtashley $  $Author: dtashley $
195  $Date: 2002/12/01 21:29:08 $  $Date: 2002/12/01 21:29:08 $
196  \end{verbatim}  \end{verbatim}
197  \end{tiny}  \end{tiny}
198  \noindent\rule[0.25in]{\textwidth}{1pt}  \noindent\rule[0.25in]{\textwidth}{1pt}
199  \end{figure}  \end{figure}
200    
201  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
202  % $Log: c_mtn0.tex,v $  % $Log: c_mtn0.tex,v $
203  % Revision 1.4  2002/12/01 21:29:08  dtashley  % Revision 1.4  2002/12/01 21:29:08  dtashley
204  % Safety checkin.  % Safety checkin.
205  %  %
206  % Revision 1.3  2002/07/31 04:37:50  dtashley  % Revision 1.3  2002/07/31 04:37:50  dtashley
207  % Number theory chapter title changed, some material added.  % Number theory chapter title changed, some material added.
208  %  %
209  % Revision 1.2  2001/07/01 19:43:13  dtashley  % Revision 1.2  2001/07/01 19:43:13  dtashley
210  % Move out of binary mode for use with CVS.  % Move out of binary mode for use with CVS.
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213  % $History: c_mtn0.tex $  % $History: c_mtn0.tex $
214  %  %
215  % *****************  Version 3  *****************  % *****************  Version 3  *****************
216  % User: Dashley1     Date: 12/22/00   Time: 12:56a  % User: Dashley1     Date: 12/22/00   Time: 12:56a
217  % Updated in $/uC Software Multi-Volume Book (A)/Chapter, MTN0, Miscellaneous Topics From Number Theory  % Updated in $/uC Software Multi-Volume Book (A)/Chapter, MTN0, Miscellaneous Topics From Number Theory
218  % Tcl automated method of build refined.  % Tcl automated method of build refined.
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220  % *****************  Version 2  *****************  % *****************  Version 2  *****************
221  % User: David T. Ashley Date: 7/29/00    Time: 11:49p  % User: David T. Ashley Date: 7/29/00    Time: 11:49p
222  % Updated in $/uC Software Multi-Volume Book (A)/Chapter, MTN0, Miscellaneous Topics From Number Theory  % Updated in $/uC Software Multi-Volume Book (A)/Chapter, MTN0, Miscellaneous Topics From Number Theory
223  % Edits, addition of solutions manual volume.  % Edits, addition of solutions manual volume.
224  %  %
225  % *****************  Version 1  *****************  % *****************  Version 1  *****************
226  % User: David T. Ashley Date: 7/29/00    Time: 9:34p  % User: David T. Ashley Date: 7/29/00    Time: 9:34p
227  % Created in $/uC Software Multi-Volume Book (A)/Chapter, MTN0, Miscellaneous Topics From Number Theory  % Created in $/uC Software Multi-Volume Book (A)/Chapter, MTN0, Miscellaneous Topics From Number Theory
228  % Initial check-in.  % Initial check-in.
229  %  %
230  %End of file C_MTN0.TEX  %End of file C_MTN0.TEX

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