- #1

- 394

- 81

## Homework Statement

The book wants me to use direct proof.

if p is a prime and k is an integer for which 0 < k < p, then p divides ##\left( \frac p k \right)##

## Homework Equations

##\left( \frac p k \right) = \frac {p!} {k!(p-k)!}##

## The Attempt at a Solution

the fraction line in ##\left( \frac p k \right)## isn't supposed to be there, but i didn't know how to remove it..its supposed to be p choose k.

proof. Suppose p is prime and k ##\epsilon## Z such that 0 < k < p.

##p! = \left( \frac p k \right)k!(p-k)!##

##p!## has a factor of p.

##k!## has no factor of p because p > k.

##(p-k)!## has no factor of p because p > (p-k).

So ##\left( \frac p k \right)k!(p-k)! = p\frac {(p-1)!} {k!(p-k)!} k!(p-k)!##

and ##p## must be a factor of ##\left( \frac p k \right)## since ##p## is not a factor of ##k!## nor ##(p-k)!## but how do you show that ##\frac {(p-1)!} {k!(p-k)!}## is an integer?