Factorial Divisible by Binary Root

Theorem

Let $n \in \Z: n \ge 1$.

Let $n$ be expressed in binary notation:

$n = 2^{e_1} + 2^{e_2} + \cdots + 2^{e_r}$

where $e_1 > e_2 > \cdots > e_r \ge 0$.

Let $n!$ be the factorial of $n$.

Then $n!$ is divisible by $2^{n - r}$, but not by $2^{n - r + 1}$.

Proof

A direct application of Factorial Divisible by Prime Power.

$\blacksquare$