De Polignac's Formula/Examples/2 in 1000

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:

$2^{995} \nmid 1000!$

Let $\mu$ denote the power of $2$ which divides $1000!$

$\ds \mu$

$=$

$\ds \sum_{k \mathop > 0} \floor {\frac {1000} {2^k} }$

$\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 $

$=$

$\ds 994$

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