Let $m >1$ and $S=\{r_1, r_2, \ldots, r_{φ(m)}\}$ as in the proof of Euler’s theorem. Prove that the product $r_1·r_2·\ldots \cdot r_{φ(m)}≡−1 \pmod m$.(Use ideas similar to the proof of Wilson’s Theorem. This conclusion could help with the last line of the proof of Euler’s theorem.)

Can anyone help me with this even with the hint? I'm kind of stumped.

  • $\begingroup$ See here $\endgroup$ – J. W. Tanner Mar 8 '19 at 2:54
  • $\begingroup$ I've added some MathJax/$\LaTeX$ syntax to format your math expressions (see here for a brief introduction). It seems evident that the first part of your Question's body was taken from a math text, since it refers to a prior pair of theorems. So I quoted that, but you should recognize the context for this problem is available to you but not your Readers. You should therefore strive to make up this deficit by providing some context of your own (what motivates the problem, why is it difficult or interesting, what did you try, etc.). $\endgroup$ – hardmath Mar 8 '19 at 3:09

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