Loading [MathJax]/extensions/TeX/mathchoice.js

Tuesday, August 10, 2021

2021/059) Let p be a prime number. Find the fractional part of \dfrac{(p+1)!}{p^2}.

 Because p is prime by wilson theorem p | (p-1)!+1

Or p^2| (p! + p)

or p^2| (p! + p) (p+1)

Or  p^2| (p+1)! + p(p+1)

or (p+1)!  \equiv  -p(p+1) \pmod p^2

So fractional part of \frac{(p+1)!}{p^2} is same as fractional part of \frac{-p(p+1)}{p^2} or is \frac{p-1}{p}

No comments: