Determine the value of:
{(2021)!/(2017)2}
Where, {x} = x - ⌊x⌋
Note: ⌊x⌋ denotes the floor function - that is, the greatest integer less than or equal to x.
*** Computer program assisted solutions are welcome, but an analytic solution is preferred.
The quotient can be rewritten as (2021*2020*2019*2018)*(2016)!/2017.
{(2021*2020*2019*2018)/2017)}={(4*3*2*1)/2017}=24/2017
So the real question is what about (2016)!/2017 ?
I played with smaller values of this: (n-1)!/n for smaller values of n and was able to conjecture the answer would be 0 if n is composite (except 4) and n-1 if n is prime.
It turns out this conjecture was correct!
https://en.wikipedia.org/wiki/Wilson's_theorem
Since 2017 is prime we have {2016!/2017}=2016/2017
The solution to the puzzle is then
(2017-24)/2017=1993/2017
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Posted by Jer
on 2023-03-25 11:20:04 |
solution
