Pascal's Triangle and Its Relatives

Exercises

Other mods

8. (mod 15) Shading the boxes of Pascal's triangle with numbers congruent to

0 (mod 15)	1, ... 13, or 14 (mod 15)	1, 2, 4, 5, 7, 8, 10,11, 13, or 14 (mod 15)	1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, or 14 (mod 15)

Observe Z15 has two proper subgroups,

H1 = {0,5,10}, ispmorphic to Z3, and

H2 = {0,3,6,9,12}, ispmorphic to Z5.

Then Z15/H2 is isomorphic to Z3 and the union of coset 1 + H2 and 1 + H2 corresponds to 1 and 2 in Z3. This explains the third picture.

Also, Z15/H1 is isomorphic to Z5 and the union of cosets 1 + H1, 2 + H1, 3 + H1 and 4 + H1 corresponds to {1,2,3,4} in Z5. This explains the fourth picture.

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