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{\Huge 189-235A:  Algebra 1} \\  \sk
{\Huge Practice Midterm Exam}  \\   
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\def\Z{{\bf Z}}
\def\C{{\bf C}}
\def\R{{\bf R}}
\def\Q{{\bf Q}}


\sk\noindent
1. Recall that the {\em power set} of a set $A$, denoted $P(A)$, is the
set of all subsets of $A$. Let $f:A\rightarrow P(A)$ be a function from
$A$ to its power set.
Show that the subset $B$ of $A$ defined by
$$ B = \{ a\in A \mbox{ such that } a\notin f(a) \}$$
is not an element of the image of $f$.

\vskip 2.5in



\sk\noindent
2. Compute the greatest common divisor of the integers
\begin{eqnarray*}
a & = & 13200000008150000000000000000132,  \\
b & = & 1320000000815. 
\end{eqnarray*}

\sk\noindent
3.  Compute the least residue modulo $N=95$ of the integer
$$ 3^{11000000000000000000000000000000000000000000}  $$


\sk\noindent
4. Using only the  basic properties of the $\gcd$ proved in class, 
(and {\em not} the fundamental theorem of arithmetic)
show that if a $p$ is a prime and $p$ divides a product $ab$ of
two integers, then $p$ necessarily divides either $a$ or $b$.















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