Question

How to solve this?If a `in ZZ_8`is an invertible element,demonstrate that `hat2x=a` don't have solutions `x in ZZ_8`

Answer 1

See explanation...

Explanation:

If there is an `x` such that:

`hat(2)x = a`

then we find:

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

Then if `a` had an inverse `a^(-1)`, we would find:

`hat(0) = hat(0)a^(-1) = hat(4)a*a^(-1) = hat(4)*hat(1) = hat(4)" "` which is false.

So there can be no such inverse. That is `a` is not invertable.