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Let $\alpha$, $1 \lt \alpha \lt \varphi(n)$, $\gcd(\alpha, \varphi(n)) = 1$, and $\beta \equiv \alpha^{-1} \pmod {\varphi(n)}$, where $\varphi$ is Euler's totient function.

When n is even, how would I prove:

$$(n-1) \equiv (n - 1)^{\beta} \pmod n$$

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    you already got an answer in your previous question - it's the *parity* of $\beta$ who decides2011-04-07
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    Instead of asking how would I prove, it would be better to say what you have tried.2011-04-07
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    I have retracted the -1, this is about showing $\beta$ is odd, which is a different problem. Unfortunately cannot retract the close vote. Apologies.2011-04-07

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