Consider the set Z5 0 1 2 3 4 with addition and multiplicat

Consider the set Z_5 = {0, 1, 2, 3, 4} with addition and multiplication given modulo 5 (i.e., declaring 5 to be equivalent to zero, so that for example 2 + 4 = 1 and 3.4 = 2) Find all additive and multiplicative inverses.

Solution

For Z₅, additive identity = 0, since 0 + n = n + 0 = n for all n.

Multiplicative identity = 1, since 1.n = n.1 = n

Additive inverses of elements of Z_5:

    Element                     Inverse

     1                                4

      2                               3

      3                               2

       4                              1

REASON: Example: 1 + 4 = 5. Under modulo 5, RHS is 0. Similarly other elements.

Multiplicative inverses of elements of Z₅:

Element                  Inverse

1                               1

2                               3

3                               2

4                                4

REASON: 1.1 =1.

2.3 = 6. Under modulo 5, RHS becomes 1. Similarly other elements