All the binomial coefficients $binom{n}{ i}$ are divisible by a prime $p$ only if $n$ is a power of $p$.

I'm looking for a "high school / undergraduate" demonstration for the:

All the binomial coefficients $\binom{n}{i}=\frac{n!}{i!\cdot (n-i)!}$ for all i, $0\lt i \lt n$, are divisible by a prime $p$ only if $n$ is a power of $p.$

There is a "elementary" (good for high school) 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.