Question

How to solve this? If `b in ZZ_8` is an non-invertible element,demonstrate that `hat2x=b` have exactly two solutions `x in ZZ_8`.

Answer 1

See explanation...

Explanation:

If `b` is an even element of `ZZ_8` then there is at least one `x` such that:

`hat(2)x = b`

Then:

`hat(2)(x+hat(4)) = hat(2)x+hat(2)*hat(4) = b+hat(8) = b+hat(0) = b`

So there are at least two solutions of:

`hat(2)x = b`

Since there are two solutions for each of the four even elements of `ZZ_8`, there can be no more than two solutions for any one of them.

We also find:

`hat(4)*b = hat(4)*hat(2)x = hat(8)x = hat(0)x = hat(0)`

So `b` is a zero divisor and as a result, non-invertible.

Conversely, just checking each of the odd elements of `ZZ_8`, we find:

`hat(1)*hat(1) = hat(1)`

`hat(3)*hat(3) = hat(9) = hat(1)`

`hat(5)*hat(5) = hat(25) = hat(1)`

`hat(7)*hat(7) = hat(49) = hat(1)`

So these elements are all self inverse.

So every non-invertable element `b` is even and has exactly two solutions to:

`hat(2)x = b`