Let $n$ be a non-zero natural integer and let $p$ be a prime number. Justify the formula $\nu_{p}(n!) = \sum_{k=1}^{+\infty} E\left(\frac{n}{p^{k}}\right)$ and show that $$\frac{n}{p} - 1 < \nu_{p}(n!) \leqslant \frac{n}{p} + \frac{n}{p(p-1)}$$
Let $n$ be a non-zero natural integer and let $p$ be a prime number. Justify the formula $\nu_{p}(n!) = \sum_{k=1}^{+\infty} E\left(\frac{n}{p^{k}}\right)$ and show that
$$\frac{n}{p} - 1 < \nu_{p}(n!) \leqslant \frac{n}{p} + \frac{n}{p(p-1)}$$