Final answer:
To build the addition, multiplication, and power tables mod 13, perform these operations on numbers from 0 to 12. The primitive roots mod 13 are 2, 6, 7, and 11. Z13 is a cyclic group under addition modulo 13.
Step-by-step explanation:
To build the addition, multiplication, and power tables mod 13, we need to perform these operations on numbers from 0 to 12 (mod 13). For addition, we simply add the numbers modulo 13. For example, 5 + 8 mod 13 = 3. For multiplication, we multiply the numbers modulo 13. For example, 5 x 8 mod 13 = 9. For powers, we raise the numbers to a given power modulo 13. For example, 5^3 mod 13 = 8.
A primitive root mod 13 is a number that generates all the numbers from 1 to 12 when raised to different powers modulo 13. In this case, the primitive roots mod 13 are 2, 6, 7, and 11.
Z13 is the group of integers modulo 13, which consists of the numbers 0 to 12. It forms a cyclic group under addition modulo 13, meaning we can generate all the numbers in Z13 by repeatedly adding a generator (primitive root) to itself. Therefore, Z13 is cyclic.