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{\bf{\centerline{ Math 431 Problem Set 3}}}


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{\bf 1.} Ex 5.2\smallskip

{\bf2.} Ex 7.1\smallskip 

{\bf 3.} Ex 7.2   
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\smallskip  {\bf{4.}} Ex 8.1\bigskip

{\bf Numerical Problems}\smallskip

 {\bf 5.} Ex 8.2 \smallskip

{\bf 6.} Ex 8.3 \smallskip

{\bf 7.} Ex 8.4\bigskip


{\bf Exploration: Irrational numbers and continued fractions}
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{\bf 8.} Try to extend the process in Problem Set 2, 11(b) to find the
beginnings  of infinite
 simple
continued fraction expansions for irrational numbers. For example,
 $$\pi=3+{1\over{7+{1\over{15+{1\over \ldots}}}}}$$
Calculate more terms in the continued fraction expansion of $\pi$.
Calculate the start of the  continued fraction expansions for $e, e^2,e^3, \sqrt
2,
\sqrt 3, \root 3 \of 2$. Any conjectures?\smallskip
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{\bf Exploration: Simultaneous Linear Diophantine Equations}\medskip

{\bf 9.} Try to determine if the  following systems of simultaneous lienar
Diophantine equations have a solution for integers $x$, $y$, $z$. If so, try to
find a solution and to describe all solutions of the equations. It may be
helpful to consider also the problem of finding solutions to the equations in
the real numbers. 



(a) $$\left\{ 
      \eqalign{4x+6y&=10\cr
          6x+9y&=15\cr }\right\}$$

(b)   $$\left\{ 
      \eqalign{4x+6y&=10\cr
          6x+9y&=20\cr }\right\}$$

(c) $$22x+ 33y+6z=5$$

(d) $$21x+6y+12z=4$$

(e) $$\left\{ 
      \eqalign{2x+3y-5z&=-3\cr
          3x+2y+4z&=7\cr
         4x-4y-2z&=2}\right\}$$

(f)  $$\left\{ 
      \eqalign{2x+3y-5z&=-3\cr
          3x+2y+4z&=7\cr}\right\}$$

Can you find a sytematic proceedure for determining all solutions in integers of
any such system in any number of variables? 







%{\bf Ingenuity} \smallskip


 
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