Let $p$ be a prime number. Show that, for all $n \in \mathbb { N }$, $$v _ { p } ( n ! ) = \sum _ { k = 1 } ^ { + \infty } \left\lfloor \frac { n } { p ^ { k } } \right\rfloor$$
Let $p$ be a prime number. Show that, for all $n \in \mathbb { N }$,
$$v _ { p } ( n ! ) = \sum _ { k = 1 } ^ { + \infty } \left\lfloor \frac { n } { p ^ { k } } \right\rfloor$$