Pascal's Triangle and Its Relatives

Exercises

Other mods

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

0 (mod 8)	1, ... 6, or 7 (mod 8)	1, 3, 5, or 7 (mod 8)	1, 2, 3, 5, 6, or 7 (mod 8)

Observe Z8 has two proper subgroups,

H1 = {0,4}, ispmorphic to Z2, and

H2 = {0,2,4,6}, ispmorphic to Z4.

Then Z8/H2 is isomorphic to Z2 and the coset 1 + H2 corresponds to 1 in Z2. This explains the third picture.

Also, Z8/H1 is isomorphic to Z4 and the union of cosets 1 + H1, 2 + H1, and 3 + H1 corresponds to {1,2,3} in Z4. This explains the fourth picture.

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