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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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.