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\begin{document}

\title{Say\i lar Kuram\i na Giri\c s}
\subtitle{Al\i\c st\i rmalar V}
\author{David Pierce}
\date{21 Aral\i k 2017}
\publishers{%Matematik B\"ol\"um\"u, MSGS\"U\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}

\maketitle
\thispagestyle{empty}

\begin{exercise}
  Hem t\"umevar\i m ile
hem kalanda\c sl\i k ile
her $n$ sayma say\i s\i\ i\c cin
a\c sa\u g\i daki b\"olebilmeleri kan\i tlay\i n.
\begin{compactenum}[(a)]
\item  
$5\divides 2^{4n}+4$,
\item
$7\divides 3^{4n+1}+4^{n+1}$.
\end{compactenum}
\end{exercise}

\begin{exercise}
\"Oklid Algoritmas\i yla,
e\u ger
\begin{gather*}
  a_1=a_2\cdot b_1+a_3,\\
a_2=a_3\cdot b_2+a_4,\\
a_3=a_4\cdot b_3+a_5,\\
\makebox[3cm]{\dotfill}\\
a_{n-1}=a_n\cdot b_n+a_{n+1},\\
a_n=a_{n+1}\cdot b_{n+1},
\end{gather*}
$a_n\neq0$, ve $\size{a_{n+1}}=d$ ise, o zaman
\begin{equation}\label{eqn:gcd}
 d=\gcd(a_1,a_2). 
\end{equation}
%Normalde $a_1>a_2>\dots>a_{n+1}>0$ ama gerekmez.  
Her durumda
\begin{equation}\label{eqn:cfrac}
  \frac{a_1}{a_2}
=b_1+\cfrac1{b_2+\cfrac1{b_3+\cfrac1{\ddots}}}.
\end{equation}
Se\c cti\u giniz $(b_1,\dots,b_{n+1})$ listesi
ve pozitif $d$ say\i s\i\ i\c cin
\"oyle $a_1$ be $a_2$ bulun ki
\eqref{eqn:gcd} ve \eqref{eqn:cfrac} e\c sitlikleri do\u gru olsun.
\"Orne\u gin $(3,1,4,1)$ ve $5$ ile $115$ ve $30$ \c c\i kar.
\end{exercise}

\begin{exercise}
$\fib n$, $n$'inci \textbf{Fibonacci say\i s\i} olsun.
\"Ozyineli tan\i ma g\"ore
\begin{align*}
  \fib 1&=1,&\fib 2&=1,&\fib{n+2}=\fib n+\fib{n+1}.
\end{align*}
A\c sa\u g\i daki e\c sitli\u gi kan\i tlayarak
her iki ard\i\c s\i k Fibonacci say\i s\i n\i n
birbirine asal oldu\u gunu g\"osterin.
\begin{equation*}
  \fib{n+3}\fib n-\fib{n+2}\fib{n+1}=(-1)^{n+1}.
\end{equation*}
\end{exercise}

\begin{exercise}\mbox{}
  \begin{compactenum}[(a)]
    \item
    Se\c cti\u giniz birbirine asal olan $k$ ve $m$ i\c cin,
    herhangi $a$ ve $b$ i\c cin
    \begin{align*}
      x&\equiv a\pmod k,&
      x&\equiv b\pmod m  
    \end{align*}
    sistemini \c c\"oz\"un.
 %   Se\c cti\u giniz $m$'n\i n \"u\c ctane basama\u g\i\ olsun.
    \"Orne\u gin $(k,m,a,b)=(17,101,3,10)$ ise
    $x\equiv717\pmod{1717}$.
    \item
    Benzer \c sekilde $n$ hem $k$'ye hem $m$'ye asal ise
    \begin{align*}
      x&\equiv a\pmod k,&
      x&\equiv b\pmod m,&
      x&\equiv c\pmod n  
    \end{align*}
    sistemini \c c\"oz\"un.
    \"Orne\u gin $(n,c)=(8,4)$ ise
    $x\equiv5868\pmod{13\,736}$.
  \end{compactenum}
\end{exercise}

\begin{exercise}
  Verilen $k$ ve $m$ i\c cin
bir mod\"ul\"u bulun ki
ona g\"ore kalanda\c sl\i k,
hem $k$'ye hem de $m$'ye g\"ore kalanda\c sl\i\u ga denk olsun.
\end{exercise}

\end{document}
