Let p be a prime. What is the remainder of (p−1)!, when divided by p? In other words: find a simple formula for (p−1)!∈Z_p (find a simple formula for the integer in {0,1,...,p−1} that is equivalent to (p−1)! modulo p). Do the computation out for small examples p=2,3,5,7, and try and spot a pattern. It might help to observe that for for p prime and for all a∈{0,1,...,p}, there is a unique x∈{0,1,...,p−1} so that ax≡1modp (though you will need to prove this fact).

Let p be a prime. What is the remainder of (p−1)!, when divided by p? In other words: find a simple formula for (p−1)!∈Z_p (find a simple formula for the integer in {0,1,...,p−1} that is equivalent to (p−1)! modulo p). Do the computation out for small examples p=2,3,5,7, and try and spot a pattern. It might help to observe that for for p prime and for all a∈{0,1,...,p}, there is a unique x∈{0,1,...,p−1} so that ax≡1modp (though you will need to prove this fact).

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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