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%TCIDATA{<META NAME="Title" CONTENT="Elementary Number Theory">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:08:11 1997}
%TCIDATA{Language=American English}
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\begin{document}


\section{\label{Prime factorization}Elementary Number Theory}

The arithmetic of positive integers exhibits many interesting properties.
Many of these properties are related to integers called primes.

\strut

\subsection{\ Prime Factorization}

A positive integer greater than $1$ is a \emph{prime} if it is not evenly
divisible by any positive integer except $1$ and itself. The list of primes
begins with $2,3,5,7,11,13,17,\ldots $. Every positive integer greater than $%
1$ can be factored into a product of powers of primes. You can identify a
prime by the fact that it equals its own prime factorization.

To factor integers into products of powers of primes in \textsl{Scientific
Notebook}, place the insertion point inside the number and choose \textsf{%
Factor}. 
\index{Factor@Factor!integer@integer}\label{Factor}

\begin{quote}
$\blacktriangleright $ \textsf{Factor}
\end{quote}

\begin{quotation}
$12345=\,3\times 5\times 823\medskip $

$4733\,64564\,31063\,80000=\allowbreak 2^{5}3^{10}5^{4}7^{3}11^{2}13\times
17\times 19\times 23\medskip $

$82723=\allowbreak 82723$
\end{quotation}

\strut

You can use \textsf{Simplify} or \textsf{Evaluate} to return any of the
preceding factorizations to integer form.

\strut

\subsection{\ Greatest Common Divisor and Least Common Multiple \label%
{Greatest common divisor}}

The \emph{greatest common divisor} 
\index{Greatest common divisor@Greatest common divisor} of a collection of
integers 
\index{Gcd@gcd}\label{gcd, integers}is the largest integer that evenly
divides every integer in the collection.

\strut

To find the greatest common divisor 
\index{Greatest common divisor@Greatest common divisor} of a collection of
integers,

\begin{itemize}
\item  Type $gcd$ in mathematics. (The name $\gcd $ should turn gray when
you type the $d$.)

\item  Enclose the list of numbers, separated by red commas, in brackets.

\item  Leave the insertion point in the list, and

\begin{itemize}
\item  Click 
\FRAME{itbpF}{0.3001in}{0.3001in}{0.0701in}{}{}{evaluate.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.3001in;height 0.3001in;depth
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"0";croptop "1";cropright "1";cropbottom "0";filename
'Evaluate.wmf';file-properties "XNPEU";}}

- or -\medskip

\item  Choose \textsf{Evaluate}.
\end{itemize}
\end{itemize}

\strut

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\gcd (35,15,65)=\,5$

$\gcd \left( 910,2405,5850,2665\right) =\,65\smallskip $

$\gcd \left( 104,221\right) =\,13$
\end{quotation}

\strut

\begin{quotation}
\emph{Note\quad }If you enter the function $\gcd $ from the keyboard while
in mathematics mode, the $gc$ appears in red italics until you type the $d$,
then the function name $gcd$ changes to a gray, nonitalic $\gcd $. \textsl{%
Scientific Notebook} substitutes the function $\gcd $ for the three-letter
sequence $g$, $c$, and $d$. You can also choose $\gcd $ from the dialog that
appears when you click \FRAME{itbpF}{0.2992in}{0.3001in}{0.0692in}{}{}{%
function.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid-file "F";width
0.2992in;height 0.3001in;depth 0.0692in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1.0246";cropright
"0.9850";cropbottom "0.0337";filename 'FUNCTION.WMF';file-properties
"XNPEU";}} or choose \textsf{Insert + Math Name}.
\end{quotation}

\strut

\label{Least common multiple}The \emph{least common multiple} of a
collection of integers 
\index{Gcd@gcd}is the smallest positive integer that is evenly divisible by
every integer in the collection. To find the least common multiple of a
collection of integers, evaluate the function $%
\limfunc{lcm}$ applied to the list of numbers enclosed in brackets and
separated by commas. Leave the insertion point in the list and choose 
\textsf{Evaluate}.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\limfunc{lcm}\left( 24,36\right) =\,72\smallskip $

$\limfunc{lcm}\left( 35,15,65\right) =1365$
\end{quotation}

\strut

You can enter the function $\limfunc{lcm}$ from the keyboard while in
mathematics mode. It changes to gray, nonitalic letters on your screen. (If
it does not appear on the function list under \FRAME{itbpF}{0.2992in}{%
0.3001in}{0.0692in}{}{}{function.wmf}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid-file
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"0.9850";cropbottom "0.0337";filename 'FUNCTION.WMF';file-properties
"XNPEU";}}, you can add it to the list{}.)

You can also determine both the greatest common divisor and least common
multiple by inspection after applying \textsf{Factor} to each of the numbers
in the list.

\subsection{\ Factorials \label{Factorial}}

Factorial 
\index{Factorial@Factorial}is the function of a nonnegative integer $n$
denoted by 
\[
n! 
\]
and defined for positive $n$ by the product 
\[
n!=1\times 2\times 3\times 4\times \cdots \times n 
\]
and for zero by 
\[
0!=1 
\]
You can compute factorials with \textsf{Evaluate}.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$3!=6$

$7!=5040$

$10!=3628800$
\end{quotation}

\strut

\subsection{\ Related topics}

\begin{itemize}
\item  
\hyperref{Integer Solutions}{}{}{DM12-1.tex#Integer solutions}

\item  \hyperref{Congruence Modulo m}{}{}{DM12-2.tex#Congruence}

\item  \hyperref{Generating Large Primes}{}{}{DM12-2.tex#Large primes}

\item  \hyperref{Greatest and Smallest Integer Functions}{}{}{DM2-4.tex#Floor%
}
\end{itemize}

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"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Factor}{}{}{%
DMIndex.tex#Factor}, \hyperref{Factorial}{}{}{DMIndex.tex#Factorial}, 
\hyperref{$\gcd $}{}{}{DMIndex.tex#Greatest common divisor}, \hyperref{%
Greatest common divisor}{}{}{DMIndex.tex#Greatest common divisor}, \hyperref{%
Least common multiple}{}{}{DMIndex.tex#Least common multiple}, \hyperref{%
Prime factorization}{}{}{DMIndex.tex#Prime factorization}}

\end{document}
