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\rightline{\sevenrm March 19, 2000}

\centerline{\largestbold Binary homework} 
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\centerline{\bf Due Monday, March 27}
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1.~Write 13 and 38 in binary. Compute in binary the sum and product
of 13 and 38. Convert the results (the two binary answers) back to 
decimal. Are the answers correct? Show your work, please!

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The following table may be useful.

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\centerline{
\vbox{
\halign{\hfil$ #$\hfil \tabskip =3pt
&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil
&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&\hfil#\hfil&
\hfil#\hfil&\hfil#\hfil\cr
n= & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\cr
%\everycr{\noalign{\smallskip\hrule\smallskip}}
2^n\!= & 1 & 2 & 4 & 8 & 16 & 32 & 64 & 128 & 256 & 512 & 1024 & 2048 & 4096\cr}
}}
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2.~Consider this pattern of 0's and 1's: 111001 (three ones
followed by two zeros followed by one one). Suppose that $N$ is a
number which when written in binary is some (unknown) number of
repetitions of this pattern. For example, if $N$ were three repetitions
of the pattern (in binary, 111001111001111001\footnote*
{$\overbrace{111001}\overbrace{111001}\overbrace{111001}$}), the number
$N$ would be (in decimal) 237,177. Verify that \underbar{no} number
produced in such a way can be a prime number.

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{\bf Hint} All you need to do is show that $N$ can be written as a
product of two other numbers -- take advantage of the binary form. You
do \underbar{not} need to do extensive computation, but you must give
me some reasonable explanation of your assertion. I don't want only
verification that the specific $N$ above is not prime. I want you to
explain why any $N$ fitting the description above can't be prime. This
is a problem which can be answered with a bit of thought more easily
than with lots of computation.

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3.~Suppose that $W$ is some really enormous number which when written
in standard decimal form using a standard font size (like that used in
this problem statement) has a length of {\bf 5 feet}. Give some
reasonable over- and under-estimates for how long $W$ would be when
written in binary form using the same font size. Your answer need not
be precise, but produce some estimates for the lengths together
with a few sentences justifying them. I am more interested in your
justification than I am in any specific quantitative answer!

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