De Polignac's Formula/Examples/2 in 720 Factorial

Example of Use of De Polignac's Formula

The prime factor $2$ appears in $720!$ to the power of $716$.

That is:

$2^{716} \divides 720!$

but:

$2^{717} \nmid 720!$

Proof

Let $\mu$ denote the power of $2$ which divides $720!$

\(\ds \mu\)

\(=\)

\(\ds \sum_{k \mathop > 0} \floor {\frac {720} {2^k} }\)

De Polignac's Formula

\(\ds \)

\(=\)

\(\ds \floor {\frac {720} 2} + \floor {\frac {720} 4} + \floor {\frac {720} 8} + \floor {\frac {720} {16} } + \floor {\frac {720} {32} }\)

\(\ds \)

\(\)

\(\ds + \floor {\frac {720} {64} } + \floor {\frac {720} {128} } + \floor {\frac {720} {256} } + \floor {\frac {720} {512} }\)

\(\ds \)

\(=\)

\(\ds 360 + 180 + 90 + 45 + 22 + 11 + 5 + 2 + 1\)

\(\ds \)

\(=\)

\(\ds 716\)

$\blacksquare$