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

All the binomial coefficients $\binom{n}{i}$ where $0\lt i \lt n$, are divisible by a primeponly ifnis a power ofp.

There is a "elementary" demonstration of reciprocal here: Binomial coefficient

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.