Fermat's theorem

1 6 ≡ 14 (mod 4) and 3 ≡ -1 (mod 4). Is it true that 6 • 3 ≡14(-1) (mod 4)?

If a ≡ b (mod 4) and c ≡ d (mod 4), can you prove that ac ≡ bd (mod 4)?

2 Make a multiplication table modulo 4 for the complete set of residues ﹛-4, 5,2, -1﹜. Does it have the same structure as the table you found for q 1.14?

3 If a ≡ b (mod n) and c ≡ d (mod n), use the equality ac-bd≡ (a-b)c + b(c-d) to prove that ac ≡ bd (mod n).

4 If a1, …, an is a complete set of residues modulo n, and the product modulo n of ai and aj is defined to be ak where aiaj (mod n), is ak uniquely defined?

Is there an element az such that a-za≡az for all i?

Is there an element aI such that aIai ≡ ai for all i?

5 If a1 …, an is a complete set of residues modulo n, and bu • • •, bn is a complete set of residues modulo n, how would you establish that the tables of products modulo n obtained from these two sets have the same structure? Your answer establishes a well-defined table for (Zn, x).

6 Examine the tables for multiplication modulo 3, 4, …, 16 given in table 3.1. Is there an identity in (Zn, x), for every n?

7 Compare the second row of the tables for (Z3, x), (Z4, x), …, (Z1o, x) with your answers to q 2.28. What is the similarity?

8 Compare the third row of the tables for (Z3, x), …, (Z10, x) with your answers to q 2.29. What is the similarity?

9 Compare the fourth row of the tables for (Z3, x), …, (Z10, x) with your answers to q 2.30. What is the similarity?