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\def\mod{{\ \rm mod\ }}
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\rightline{\sevenrm April 20, 2000}\smallskip
\centerline{\largerbold Math review for the final exam}

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\noindent {\bf Exam conditions} The final exam will be ``open book,
open notes''. You may bring to the exam anything additional to help
you which is {\sl non-living}.\footnote*{\sevenrm
\baselineskip=.7\baselineskip The instructor will judge if something
is ``alive''. I included this stipulation because of the following
story: At Caltech one semester, a physics instructor said that on the
final exam, students could use ``anything they could carry in.'' One
student carried in Richard Feynman. You should know who Richard
Feynman is (ground-breaking theoretical physicist) and something of
his character (first-class crazy person) to appreciate this.\par}

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\noindent {\bf Rules} 

\item{$\bullet$} In each case, you should show your work, explain your
answer, etc.: an answer alone generally will {\it not}\/ be sufficient
for full credit!

\item{$\bullet$} Write answers to ``mod'' questions in standard form:
although $20= 2 = -16\mod 18$ etc., the standard form would be {\bf 2}:
always an integer between 0 and the modulus.

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\prob Which of $\left(10^{44}\right)^{55}$ and
$\left(10^{44}\right)\!\cdot\! \left(10^{55}\right)$ is larger?

\prob What is $56 \mod 6$? 

\prob What is $3798107541 \mod 9$? 

\prob What is $3798107541 \mod 11$?
 
\prob What is $7^{322222224512} \mod 8$? ($7=8-1$.)

\prob What is $7^{322222224512} \mod 7$? 

\prob What is $7^{322222224512} \mod 6$? ($7=6+1$). 

\prob What is $3\cdot (6^{200})+ 2\cdot (8^{333})+4 \mod 7$?
($6=\ldots$ and $8=\ldots$.)

\prob What is $42^{1234567} \mod 4$?

\prob What is the table of addition mod 5?

\prob What is the table of multiplication mod 5?

\prob Solve $2x+1=4 \mod 5$.

\prob Solve $2x+1=4 \mod 6$.

\prob Solve $3x+1=4 \mod 6$.

\prob Define {\bf prime number}. Explain briefly why arithmetic mod a
prime number is more like standard arithmetic than arithmetic mod a
non-prime number. 

\prob What is 23 in binary?

\prob What is 10110001 in decimal?

\prob How many different bitstrings 7 bits long are there? 

\prob What is the xor of the bitstrings $11001\,11100$ and $10101\;00111$?

\prob Which is more work (``work''$=$one-digit arithmetic operations):
squaring one hundred numbers 50 decimal digits long or adding ten
pairs of numbers 5,000 decimal digits long?  




\eject\end
