Problem with modulo in field

Question

I have problem with comprehending how works number in field when it's rasied to negative power.

For instance if we have $4^{-1}$ at $Z_{5}$ I tried to write it as $4\cdot 4^{-1}+4^{-1}=4^{-1}(1+4)$ so it's $4^{-1}= 0 \space mod \space5 $ ??

Answer 1

By definition $4.4^{-1}\equiv1$ mod $5$ so $4.4^{-1}+4^{-1}\equiv1+4^{-1}$ mod $5$.

Also $4.4^{-1}+4^{-1}\equiv\left(4+1\right)4^{-1}\equiv0$ mod $5$

Proved is now $1+4^{-1}\equiv0$ mod $5$ or equivalently $4^{-1}\equiv-1$ mod $5$ or equivalently $4^{-1}\equiv4$ mod $5$

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To 'find' $4^{-1}$ in situations like this just go on search for the unique element $r\in\mathbb Z_5$ that satisfies $r\times 4\equiv1$ mod $5$.