Legendre's Formula

Legendre's Formula states that

$e_p(n)=\sum_{i\geq 1} \left\lfloor \dfrac{n}{p^i}\right\rfloor =\frac{n-S_{p}(n)}{p-1}$

where $p$ is a prime and $e_p(n)$ is the exponent of $p$ in the prime factorization of $n!$ and $S_p(n)$ is the sum of the digits of $n$ when written in base $p$ .

Proof

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Legendre's Formula

Proof