Assignment:

(1)  List all the additive subgroups of Z12.

                 { 0,1,2,3,4,5,6,7,8,9,10,11 }
                              /\
                             /  \
                            /    \
                           /      \
                          /        \
                         /          \
                        /            \
                       /              \
                      /                \
                     /                  \
                    /                    \
                   /                      \
                  /                        \
          { 0,2,4,6,8,10 }             { 0,3,6,9 }    
                 |\                        /
                 | \                      /
                 |  \                    /
                 |   \                  /
                 |    \                /
                 |     \              /
                 |      \            /
                 |       \          /
                 |        \        /     
                 |         \      /     
                 |          \    /     
                 |           \  /     
                 |            \/     
             { 0,4,8 }       {0,6}
                  \           /                
                   \         /              
                    \       /              
                     \     /              
                      \   /              
                       \ /              
                        V             
                      { 0 }           


(2)  List all the additive subgroups of Z19

    {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
                              |
                              |
                              |
                              |
                            { 0 }

(3)  How many additive subgroups are there of Z100.

     2^2 5^2  should be 3^2 = 9.

1,2,4,5,10,20,15,50,100  There are 9 divisors of 100.

(4)  Find integers x and y such that

         x 13 + y 17 = 1. 

      1
13 | 17
     13
     --    3
      4 | 13
          12
          --   4
           1 | 4
               4
              ---
               0

1  3  4
1  1  4 17
0  1  3 13

        4*13 - 3*17 = 1