Legendre's Theorem/Corollary

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $B$ be the binary representation of $n$.

Let $r$ be the number of unit digits in $B$.

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

Then $2^{n - r}$ is a divisor of $n!$, but $2^{n - r + 1}$ is not.

Proof
$n$ can be represented as:

where all of $e_1, e_2, \ldots, e_r$ are integers.

Using De Polignac's Formula, we may extract all the powers of $2$ from $n!$


 * $\mu = \displaystyle \sum_{k \mathop > 0} \left\lfloor{\dfrac n {2^k} }\right\rfloor$

where $\mu$ is the multiplicity of $2$ in $n!$:
 * $2^\mu \mathrel \backslash n!$
 * $2^{\mu + 1} \nmid n!$

From Legendre's Theorem, we have:
 * $\mu = \dfrac {n - r} {2 - 1}$

Hence the result.