# Thread: Prove their own multiplicative inverses

1. ## Prove their own multiplicative inverses

Let p be a prime integer. Prove that [1] and [p-1] are the only elements in Zp that are their own multiplicative inverses.

2. Originally Posted by rainyice
Let p be a prime integer. Prove that [1] and [p-1] are the only elements in Zp that are their own multiplicative inverses.
Hint: If $x^2 \equiv 1 mod (p)$, then $x \equiv +1$ or $x \equiv -1$.
$U_p=\{[1], [2], \cdots ,[p-1]\}$ for a prime number p > 2 and $U_2=\{1\}$, where $U_p$ is the group of units in $\mathbb{Z}_p$.

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### write the integer which is its own multiplicative

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