1.  Find the smallest distance that can be measured with a
(unmarked) meter stick and an (unmarked) yard stick.  
1 meter = 39.37 inches.

             1
     3600 3937
          3600   10
           337 3600
               3370   1
                230 337
                    230   2
                    107 230
                        214   6
                         16 107
                             96  1
                             11 16
                                11  2
                                 5 11
                                   10 5
                                    1 5
                                      5
                                      0

   1    10     1    2    6      1    2    5
   1     1    11   12   35    222  257  736  3937
   0     1    10   11   32    203  235  673  3600

So  -736 36.00 + 673 39.37 = 0.01

So lay out 673 of the meter sticks and come back with 736 or the
yard sticks and the difference will be 0.01 inches.


2.  Given n = 11663 and r = 7 encode the message:

        I       A  M      S  I  C  K   
        9       1 13     19  9  3 11

 9 -> Mod[ 9^7, 11663] = 1139 
 1 -> Mod[ 1^7, 11663] =    1
13 -> Mod[13^7, 11663] = 1577 
19 -> Mod[19^7, 11663] = 7756 
 9 -> Mod[ 9^7, 11663] = 1139
 3 -> Mod[ 3^7, 11663] = 2187 
11 -> Mod[11^7, 11663] = 9961

3.  Break the code and decipher the message.

        7133 8088   9475 8088 1577 8147

The whole security of the RSA code is that factoring
is difficult.  If we can factor 11663 the code is
broken.  11663 = 107*109.

We have to find the inverse of r = 7 mod (p-1)(q-1) = 11448
We know 7 has an inverse mod 11448 because GCD(7,11448) = 1.
We find the inverse using the Euclidean Algorithm.

   1635 
7 11448 
  11445
  -----  2 
      3  7
         6 
        ---  3
         1   3
             3
            ---
             0

1635    2     3
   1  1635  3271 11448
   0    1     2    7

  3271 7 - 2 11448 = 1

And     3271*7  =  22897 = 1 mod 11448 

The private key to crack the code is 3271.
f[x_] := Mod[x^3271,11663]
Just to check we try the encryption of I AM SICK 
f[{1139,1,1577,7756,1139,2187,9961}] = 

{9, 1, 13, 19, 9, 3, 11}
 I  A   M   S  I  C   K

So it seems to work.
To break the code

f[{7133,8088,  9475,8088,1577,8147}] = 

{7, 15,    8, 15, 13, 5}
 G   O     H   O   M  E