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

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

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  n. (d) Each [a] n in Z n has an additive inverse [  a] n in Z n.

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

Ex 2. Let n = 4. Z 4 = {, , , }. The operation table with respect to “+” is as follow. +           +      = =       =      

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  n.

Ex 3. [4 · 9 · 15 · 59] 7 =

Ex 4. Let n = 4. Z 4 = {, , , }. The operation table with respect to “ ． ” is as follow. ·           ·       =       =      

Note:    , such that · =  in Z 4. The inverse of  is  in Z 4.

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

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

Ex 5. The elements of Z 15 that have multiplication inverses are , , , , , , , . Moreover,

Ex 6. What is the inverse of  in Z 191 ?

Ex 7. Find   1 in Z 26.

Ex 8. Find integers x, y satisfying [x] + [y] =  and [x] + [y] =  in Z 26.

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