Factorial Divisible by Binary Root
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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.
$\blacksquare$