Day 5 -- 9/17

Introduced a|b, congruences. Proved \equiv is an equivalence relation. Discussed briefly the idea of equivalence classes and well-definition of + and *. [went poorly]

Discussed inverses mod m -- introduced the relevant Diophantine equation. Proved if b shares a non-trivial factor with m, then b is not a unit.

Did an example of the magic table (and then reverse substitution). [went great]

Day 4 -- 9/15

Classified all multiplication tables for groups of size 1,2,3,4.

Looked at powers in modular arithmetic. Proved 3^4 = 1 (mod 10) via multiplication by 3 is a bijection.

Day 3 -- 9/10

Proved basic properties:
uniqueness of e
cancellation
uniqueness of inverses
(ab)^{-1} = b^{-1} a^{-1}

Examined (Z_4,+) vs. (U(5),*), wrote down their multiplication tables, and found an isomorphism between them.

Day 2 -- 9/8

Introduced D_3.
Made a multiplication table via Sodoku methods.

Day 1 -- 9/3

Introduced the notion of a group.
Did many examples including standard number systems, matrix groups, modular arithmetic.