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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {720} 3} + \floor {\frac {720} 9} + \floor {\frac {720} {27} } + \floor {\frac {720} {81} } + \floor {\frac {720} {243} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 240 + 80 + 26 + 8 + 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 356\) |
$\blacksquare$