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I'm a bit stuck on this question:

Let $p$ be a prime number. Prove that $(2p−1)(2p−2)\cdots(p + 1) \equiv−1\pmod{p}$. Hint: find the least positive residues of $2p−1,2p−2,\cdots,p + 1 \pmod{p}$.

I think the least positive residues are $\{0,1,2,\cdots,p\}$ but not entirely sure, and I don't see how it's going to help me answer the question below. Thanks, Alex

Thirdy Yabata
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Alex
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  • Are you familiar with Wilson's Theorem? – Gerry Myerson Oct 17 '16 at 08:49
  • Yes! I know I'll end up using it at the end of the proof, but I'm not sure how to get the left hand side of the congruence into the form needed for Wilson's Theorem. Side note - it's a 4 mark question so I assumed it required a fair amount of working? – Alex Oct 17 '16 at 20:39
  • As here in the 2nd dupe, your sequence is just the standard least positive system of nonzero residues shifted by $p$ so it remains a complete system of nonzero residues, so Wilson's Theorem applies, as here in the first dupe. $\ \ $ – Bill Dubuque Apr 16 '25 at 16:36

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Since you're working modulo $p$, you can subtract $p$ from each factor in your product. This will, by happy accident, give you the least positive residue of each factor. Then answer Prof. Myerson's question.

B. Goddard
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