So, during algebra class, one of the 5th-year students who was there to help us and I tried to find the proof for $$\left.p\middle|{p \choose k}\right. \forall k, 0<k<p \iff \text{p is a prime number}$$
So to do this we need Kummer's Theorem, which itself uses an alternate form of Legendre's formula. What I want to prove is the following identity: $$\sum_{i=1}^\infty\left\lfloor \frac n {p^i} \right\rfloor = \frac {n - s_p(n)}{p-1}$$
Where the brackets denote the floor function and $s_p(n)$ denotes the sum of all digits of $n$ when $n$ is written is base $p$.
The proof is in Wikipedia, but I'm afraid there are some steps I don't get.
