1. Congruence Classes Z n = {[0] n, [1] n, [2] n, …, [n - 1] n } = the set of congruence classes modulo n.

2. Theorem 1. Consider the rule given by [a] n +[b] n =[a+b] n, where, [a] n, [b] n  Z n. (a) “+” is a binary operation on Z n. (b) “+” is associative and commutative. (c) Z n has the additive identity [0] n. (d) Each [a] n in Z n has an additive inverse [  a] n in Z n.

3. Ex 1. [4 + 9 + 15 + 59] 7 =

4. Ex 2. Let n = 4. Z 4 = {[0], [1], [2], [3]}. The operation table with respect to “+” is as follow. + [0] [1] [2] [3] [0] [0] [1] [2] [3] [2] + [3] [1] [1] [2] [3] [4]=[0] = [5] [2] [2] [3] [0] [1] = [1] [3] [3] [0] [1] [2]

5. Theorem 2. Given [a]·[b] = [ab], where [a], [b]  Z n. (a) “·” is a binary operation on Z n. (b) “·” is associative and commutative. (c) Z n has the multiplicative identity [1] n.

6. Ex 3. [4 · 9 · 15 · 59] 7 =

7. Ex 4. Let n = 4. Z 4 = {[0], [1], [2], [3]}. The operation table with respect to “ ． ” is as follow. · [0] [1] [2] [3] [0] [0] [0] [0] [0] [2] · [3] [1] [0] [1] [2] [3] = [6] [2] [0] [2] [0] [2] = [2] [3] [0] [3] [2] [1]

8. Note:  [2]  [0], such that [2]·[2] = [0] in Z 4. The inverse of [3] is [3] in Z 4.

9. Theorem 3. [a] in Z n has a multiplicative inverse if and only if (a, n) = 1. Pf:

10. Cor 4. Every nonzero element of Z n has a multiplication inverse if and only if n is prime.

11. Ex 5. The elements of Z 15 that have multiplication inverses are [1], [2], [4], [7], [8], [11], [13], [14]. Moreover,

12. Ex 6. What is the inverse of [13] in Z 191 ?

13. Ex 7. Find [15]  1 in Z 26.

14. Ex 8. Find integers x, y satisfying [4][x] + [y] = [22] and [19][x] + [y] = [15] in Z 26.