De Polignac's Formula/Examples/2 in 1000
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Example of Use of De Polignac's Formula
The prime factor $2$ appears in $1000!$ to the power of $994$.
That is:
- $2^{994} \divides 1000!$
but:
- $2^{995} \nmid 1000!$
Proof
Let $\mu$ denote the power of $2$ which divides $1000!$
\(\ds \mu\) | \(=\) | \(\ds \sum_{k \mathop > 0} \floor {\frac {1000} {2^k} }\) | De Polignac's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {1000} 2} + \floor {\frac {1000} 4} + \floor {\frac {1000} 8} + \floor {\frac {1000} {16} } + \floor {\frac {1000} {32} }\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \floor {\frac {1000} {64} } + \floor {\frac {1000} {128} } + \floor {\frac {1000} {256} } + \floor {\frac {1000} {512} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 500 + 250 + 125 + 62 + 31 + 15 + 7 + 3 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 994\) |
$\blacksquare$