De Polignac's Formula/Examples/11 in 1000

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Example of Use of De Polignac's Formula

The prime factor $11$ appears in $1000!$ to the power of $98$.

That is:

$11^{98} \divides 1000!$

but:

$11^{99} \nmid 1000!$


Proof

Let $\mu$ denote the power of $11$ which divides $1000!$

\(\ds \mu\) \(=\) \(\ds \sum_{k \mathop > 0} \floor {\frac {1000} {11^k} }\) De Polignac's Formula
\(\ds \) \(=\) \(\ds \floor {\frac {1000} {11} } + \floor {\frac {1000} {121} }\)
\(\ds \) \(=\) \(\ds 90 + 8\)
\(\ds \) \(=\) \(\ds 98\)

$\blacksquare$