Remainder of 34! divided by 37 using Wilson's Theorem

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Homework Help Overview

The discussion revolves around finding the remainder of 34! when divided by 37, utilizing Wilson's Theorem as a foundational concept in number theory.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore the application of Wilson's Theorem, particularly how to manipulate factorial expressions under modular arithmetic. There are attempts to solve for the modular inverse and questions about the correctness of calculations and assumptions.

Discussion Status

The conversation is ongoing, with participants providing guidance on applying modular arithmetic and checking calculations. There is a mix of interpretations and attempts to clarify the steps involved in the problem.

Contextual Notes

Some participants express uncertainty about specific calculations and the application of modular inverses, indicating potential gaps in understanding or missing information regarding the problem setup.

duki
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Homework Statement



Find the remainder when 34! is divided by 37.

Homework Equations



Wilson's Theorem

The Attempt at a Solution



I understand that (p-1)! = (-1)(mod p) and that (p-2)! = (1)(mod p). I don't understand how to apply this to (p-3)! though.
 
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you know that (p-2)! = 1 (mod p). So (p-3)!*(p-2) = 1 (mod p). In this situation, 34!*35 = 1 (mod 37). Call 34! 'x' and then solve 35x = 1 mod 37, which has a unique solution since gcd(35,37) = 1.
 
So do you do..

1 = 35x
1 = (-2)x
1 = (-2)(-18)

R = -18 + 37 = 19 ??
 
35*19 isn't 1 mod 37. Don't you mean 1=(-2)(-19)? It's easy enough to check your answers with a quick calculation.
 
You're right. Thanks for the help.
 
I'm trying to find 33! / 37 now.

I have gotten to (-3)x = 18 (mod 37)... but I can't figure out what x is.
 

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