How do I solve for M modulo 2021 where M is the product of all numbers relatively prime to 2021 that are less than 2021. I know Wilson's Theorem but that's applicable for modulo p where p is prime. Should I use CRT by breaking 2021 into 43 and 47 but even that seems intractable to me. Any help is appreciated.

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    $\begingroup$ $2020!$ has several factors of $43$ and $47$ $\endgroup$ – lulu Jan 25 at 15:10
  • $\begingroup$ Of course the remainder is $0$. What makes you think otherwise? $\endgroup$ – lulu Jan 25 at 15:18
  • $\begingroup$ Yes it is the case why do you think it isnt? $\endgroup$ – Archis Welankar Jan 25 at 15:19
  • $\begingroup$ I have edited the problem as I spotted a mistake in definition of M $\endgroup$ – Mathronza Jan 25 at 15:25

All of these numbers will have multiplicative inverses (mod 2021), so they all cancel, with the exception of those numbers which are their own inverses. So $M$ is the product of the residues whose square is $1$.

We can find all such $x$ using the CRT to be $1, 988, 1033, 2020$. So their product $M = 1$.


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