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Let $m$ be the number of solutions of the equation $x^2 = 1$ in the ring $Z_n$, where $Φ_n$ is Euler group, $n> 2$.

  • Prove that $m$ is an even number.

Could anyone help me or give me a hint?

Thanks in advance.

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    $\begingroup$ Prove that the set of solutions can be naturally grouped into pairs in a certain way. $\endgroup$ – Erick Wong Jan 16 '16 at 16:27
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It's not clear to me why the question mentions $U(n)$ (I'm assuming by Euler group you mean group of units modulo $n$).

To see that the number of solutions to $x^2 = 1$ in $\mathbb Z_n$ is even if $n>2$ note that if $x^2 = 1$ then also $(-x)^2 = 1$ and $x \neq -x$ hence solutions to the equation occur in pairs.

Hope this helps.

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