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

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Answer 1 · 2017-04-07T21:49:25

See 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.

How to solve this?If a in ZZ_8in ZZ_8is an invertible element,demonstrate that hat2x=ahat2x=a don't have solutions x in ZZ_8x in ZZ_8