- #1
obo
- 2
- 0
Hi all, this homework problem's been driving me nuts. It seems like it's probably pretty straightforward and I'm missing something obvious, but I just can't work it out.
prove that if p is a prime number that p|B(p,m) where B(p,m) is the ordinary binomial coefficient (i.e. p Choose m) for 0 < m < p
B(p,m) = p!/m!(p-m)!
If you factor p out of the binomial coefficient, you're left with (p-1)!/m!(p-m)!, which must be an integer. Thus I need to be able to show that m!(p-m)!|(p-1)! somehow. I've monkeyed around with the expressions to try and recover a multiple of a binomial coefficient or something, but haven't been able to... but I'm getting the feeling that I'm taking the wrong approach here =/
Any hints here would be very much appreciated!
Cheers
Homework Statement
prove that if p is a prime number that p|B(p,m) where B(p,m) is the ordinary binomial coefficient (i.e. p Choose m) for 0 < m < p
Homework Equations
B(p,m) = p!/m!(p-m)!
The Attempt at a Solution
If you factor p out of the binomial coefficient, you're left with (p-1)!/m!(p-m)!, which must be an integer. Thus I need to be able to show that m!(p-m)!|(p-1)! somehow. I've monkeyed around with the expressions to try and recover a multiple of a binomial coefficient or something, but haven't been able to... but I'm getting the feeling that I'm taking the wrong approach here =/
Any hints here would be very much appreciated!
Cheers
