how do i construct the addition and multiplication tables for Z3?

Question

i think Z3 means the integers mod 3

Answer 1

Just make a table with 0, 1, and 2 on each side (since every integer leaves a remainder of 0, 1, or 2 when divided by 3) and compute each entry:

Addition:
0 + 0 = 0
0 + 1 = 1
0 + 2 = 2
1 + 0 = 1
1 + 1 = 2
1 + 2 = 0 (This is 3 in the integers, which is congruent to 0)
2 + 0 = 2
2 + 1 = 0 (This is 3 in the integers, which is congruent to 0)
2 + 2 = 1 (This is 4 in the integers, which is congruent to 1)

Multiplication:
0 x 0 = 0
0 x 1 = 0
0 x 2 = 0
1 x 0 = 0
1 x 1 = 1
1 x 2 = 2
2 x 0 = 0
2 x 1 = 1
2 x 2 = 1 (This is 4 in the integers, which is congruent to 1).