__high school / undergraduate__" demonstration for the:

All the binomial coefficients $\binom{n}{i}$ where $0\lt i \lt n$, are divisible by a prime

*p*only if

*n*is a power of

*p*.

There is a "elementary" demonstration of reciprocal here: http://mathhelpforum.com/number-theory/186439-binomial-coefficient.html

And for this part I'm trying to avoid using more advanced tools like the Theorem of Lucas or Kummer.

Thanks in advance to anyone who can help.