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.