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{\bf{\centerline{ Math 431 Exam 1: Monday Feb 28, 3:00--4:15pm}}}


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{\bf 1.}{\sl (10 points)} \smallskip 
{ (a)}	Define the greatest common divisor of two integers $m$ and   $n$.\smallskip
{(b)} Explain what is meant by saying that $m$ and $n$ are relatively prime.
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{\bf 2.} {\sl (10 points)}	Compute    the greatest common divisisor $d$ of  $147$ and  $99$ and
 use the Euclidean algorithm to represent it in the form $d=147x + 99y$
for some integers
$x$ and
$y$.\medskip







{\bf 3.} {\sl (15 points)} Find {\bf all} incongruent  solutions of the congruences \smallskip 
(a)  $35x\equiv  2  \pmod {41}$.\smallskip
(b) $x^2-1 \equiv 0 \pmod {16}$.\medskip


{\bf 4.} {\sl (10 points)}  How many incongruent solutions do each of the following congruences have? 
\smallskip	(a)         	$105x\equiv  2  \pmod {49}$.\smallskip			(b)	$105x \equiv 21  \pmod {49}$.
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{\bf 5.} {\sl (10 points)}
 Find an integer $x$ satisfying both congruences $x\equiv 5 \pmod{111}$ and $x\equiv 6 \pmod {113}$.\medskip

{\bf 6.} {\sl (10 points)}.\smallskip 
(a) State the Little Fermat Theorem.\smallskip
(b) The number 1063 is prime. Find the integer $a$ with $0\leq a< 1063$
 and $10^{2127}\equiv a\pmod{1063}$.\medskip

{\bf 7.} {\sl (10 points)}\smallskip 
(a)  Define Euler's $\phi$-function and state Euler's generalization of the Little Fermat Theorem.\smallskip 
(b)  Compute $\phi(180)$.\medskip


{\bf 8.} {\sl (15 points)} 
 The incongruent  squares modulo $7 $ (i.e. integers $y$ of the form $y\equiv x^2\pmod 7$ for some integer $x$ and 
 with $0\leq y<7$  ) are 
$0$,
$1$,
$4$ and $2$ (e.g. $2\equiv 3^2\pmod 7$). Thus, there are $4$ incongruent  squares modulo $7$.

\hskip .2in   Find all the incongruent squares modulo $p$ for (a) $p=3$, (b) $p=5$, (c) $p=11$
(and more $p$ if you need), and try to guess a formula in terms of $p$ for the number
 of incongruent squares modulo $p$ for any odd prime number
$p$.\medskip
 



{\bf 9.} {\sl (10 points)}\smallskip 
{ (a)} Define the term ``primitive  Pythagorean triple.'' 
Give  examples of three different  primitve Pythagorean triples.\smallskip
{(b )} Find a primitive Pythagorean triple $(a,b,c )$ with $a=25$.\medskip

































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