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{\Huge 189-235A:  Algebra 1} \\  \sk
{\Huge Midterm Exam}  \\   \sk

{\Large Wednesday, October 16}
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\def\Z{{\bf Z}}
\def\C{{\bf C}}
\def\R{{\bf R}}
\def\Q{{\bf Q}}
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{\em Each of the four questions below is worth 25 points. No calculators or
outside materials are allowed during the exam.  }

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1. Solve the following congruence equations  
(making sure you list {\em all} the distinct solutions in the given
ring).  You do not need to justify 
your answers.

(a) $3x = 4$ \ \  \  in \  \ \  $\Z/13\Z$.

(b) $3x=4$ \ \ \  in \ \ \   $\Z/12\Z$.

(c) $3x=9$ \  \  \  in \  \ \  $\Z/12\Z$.

(d) $x^2=x$ \  \  \ in \ \ \  $\Z/101\Z$.

(e) $x^2=-1$ \ \ \ in \ \ \  $\Z/303\Z$. 

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\noindent
2. a) Let $a$ and $b\in\Z$  be two integers whose
 only common divisors are
 $1$ and $-1$.  If $c$ is any integer,
prove that $a$ divides $bc$ if and only if 
$a$ divides $c$. 

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b) Give a counterexample to the statement of part a) if the ring $\Z$ of 
regular integers  is replaced by the ring $\Z[\sqrt{-5}]$ consisting of all
integer linear combinations of $1$ and $\sqrt{-5}$. 



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3. Let $n$ be an integer which is not divisible by $2$, $3$, $5$ or $7$.
Show that if $n < 121$, then $n$ is a prime number.


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4.  Compute the value of $2^{15}$ modulo $15$. 
Explain why your calculation gives a (somewhat
perverse\footnote{There are  even more perverse
ways of going about it: in the last
 decade of the 20th century, untold millions of dollars
were invested in the  project to factor
the number $15$ on a {\em quantum computer}.}) proof
that $15$ is not a prime number. 



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