How to solve this? We have the ring `(ZZ_8,+,*)`Demonstrate that the invertible elements of this ring are:`hat1,hat3,hat5,hat7`.

The elements of `ZZ_8` are `hat 0`, `hat 1`, `hat 2`, `hat 3`. `hat 4`, `hat 5`, `hat 6` and `hat 7`. Now:

The inverse of `hat 1` is the same `hat 1` since `hat 1` * `hat 1` = `hat 1`. Similarly:

The inverse of `hat 3` is the same `hat 3` since `hat 3` * `hat 3` = `hat 1`

The inverse of `hat 5` is the same `hat 5` since `hat 5` * `hat 5` = `hat 1`

The inverse of `hat 7` is the same `hat 7` since `hat 7` * `hat 7` = `hat 1`. So these elements are invertible.

Now, let's consider `hat 2`. The possible products are:

`hat 2` * `hat 0` = `hat 0`

`hat 2` * `hat 1` = `hat 2`

`hat 2` * `hat 2` = `hat 4`

`hat 2` * `hat 3` = `hat 6`

`hat 2` * `hat 4` = `hat 0`

`hat 2` * `hat 5` = `hat 2`

`hat 2` * `hat 6` = `hat 4`

`hat 2` * `hat 7` = `hat 6`

We can see that no product gives us `hat 1`. The same way we can prove that `hat 4` and `hat 6` have no inverses. By definition `hat 0` is not invertible.

Therefore, the only invertible elements are `hat 1`, `hat 3`, `hat 5` and `hat 7`

In general terms, invertible elements in these rings are the ones coprime with the order (8 in this case) of the ring