De Polignac's Formula/Examples/5 in 720 Factorial

Example of Use of De Polignac's Formula

The prime factor $5$ appears in $720!$ to the power of $178$.

That is:

$5^{178} \divides 720!$

but:

$5^{179} \nmid 720!$

Proof

Let $\mu$ denote the power of $5$ which divides $720!$

\(\ds \mu\)

\(=\)

\(\ds \sum_{k \mathop > 0} \floor {\frac {720} {5^k} }\)

De Polignac's Formula

\(\ds \)

\(=\)

\(\ds \floor {\frac {720} 5} + \floor {\frac {720} {25} } + \floor {\frac {720} {125} } + \floor {\frac {720} {625} }\)

\(\ds \)

\(=\)

\(\ds 144 + 28 + 5 + 1\)

\(\ds \)

\(=\)

\(\ds 178\)

$\blacksquare$