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\centerline{\bigrm Divisibility Tests}
\bigskip

Most people know an easy way to check if a number is divisible by 2, 5
or 10.  The purpose of this project is for you to discover how to check
if a number is divisible by other small numbers.  Follow the steps below
to complete the project.

\item{1.}The first step is to verify the usual test for divisibility by
         2.
 \itemitem{a.}Expand the number 5719 into its base 10 expansion.
 \itemitem{b.}To the expansion in part a,
              apply the rules to evaluate a number modulo 2.  Based on
              your observation, why is
              only the ones place important in determining if a number
              is divisible by 2?
 \itemitem{c.}Suppose that a number is written base 10 as
        $a_0+a_1\times 10 + a_2\times 10^2 +\cdots +a_n\times 10^n$.
        Show how using arithmetic modulo 2 gives the usual rule telling
        if a number is divisible by 2.

\item{2.}Repeat number 1, but this time use 5 instead of 2.

\item{3.}Repeat number 1, but this time use 10 instead of 5.

\item{4.}Now find a test for divisibility by 3 by following the steps
         below.
 \itemitem{a.}Expand the number 29412 into its base 10 representation.
       Rewrite what the number is mod 3 by replacing 10 and all its
       powers with their mod 3 equivalents, but leaving the digits in
       the number as they are.  Next simplify by reducing further mod 3
       until you determine a number between 0 and 2 inclusive which is
       equivalent to 2941 mod 3.
 \itemitem{b.}Repeat part a, using the number 49102 instead of 29412.
 \itemitem{c.}Based on your calculations in parts b and c, state a rule
       about how to check divisiblity by 3.
 \itemitem{d.}Show that your rule always works in the same way you
       verified the divisibility test for 2 in number 1 part c.

\item{5.}Repeat the process in number 4, using 9 instead of 3.

\item{6.}Repeat the process in number 4, using 11 instead of 3.

\item{7.}Using the same ideas as above state and show the divisibility
         tests for 4 and 8.

\item{8.}Summarize the divisibility tests for 2,3,4,5,6,8,9,10, and 11.
         Include examples.
         (In the case of 6, you can just check divisible by 3 and
         divisible by 2. Why?)

\item{9.}In part 8 you listed simple ways to check divisibility by all
         the number from 2 to 11 inclusive except for the number 7.  Use
         the same procedure as you used above to find a way to check
         that a number is divisible by 7.  The divisibility test for 7
         you develop will probably be a little more complicated than the
         other tests.

\bye