# Prove that in $\mathbb{Z}_p$, $[a]^{-1} = [a]$ if and only if $[a]=[1]$ or $[a]=[p-1]$

Let $p$ be a prime and a an integer. Prove that in $\mathbb{Z}_p$, $[a]^{-1} = [a]$ if and only if $[a]=[1]$ or $[a]=[p-1]$.

I greatly appreciate your help on this question!

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I guess $[a] \neq [0]$, right? – azimut Mar 11 '13 at 22:39
The question does not specify, it simply says a an integer which confuses me. – Michael Mar 11 '13 at 22:40
If $[a] = [0]$, then $[a]^{-1}$ does not exist. So you need $[a] \neq [0]$. – azimut Mar 11 '13 at 22:49

## 1 Answer

Hint: $[a]^{-1} = [a] \iff 1 = [a]^2 \iff [a]^2 - 1 = 0 \iff ([a] - 1)([a] + 1) = 0$.

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