De Polignac's Formula/Examples/3 in 720 Factorial

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Example of Use of De Polignac's Formula

The prime factor $3$ appears in $720!$ to the power of $356$.

That is:

$3^{356} \divides 720!$

but:

$3^{357} \nmid 720!$


Proof

Let $\mu$ denote the power of $3$ which divides $720!$

\(\ds \mu\) \(=\) \(\ds \sum_{k \mathop > 0} \floor {\frac {720} {3^k} }\) De Polignac's Formula
\(\ds \) \(=\) \(\ds \floor {\frac {720} 3} + \floor {\frac {720} 9} + \floor {\frac {720} {27} } + \floor {\frac {720} {81} } + \floor {\frac {720} {243} }\)
\(\ds \) \(=\) \(\ds 240 + 80 + 26 + 8 + 2\)
\(\ds \) \(=\) \(\ds 356\)

$\blacksquare$