De Polignac's Formula/Examples/5 in 1000

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Example of Use of De Polignac's Formula

The prime factor $5$ appears in $1000!$ to the power of $249$.

That is:

$5^{249} \divides 1000!$

but:

$5^{250} \nmid 1000!$


Proof

Let $\mu$ denote the power of $5$ which divides $1000!$

\(\ds \mu\) \(=\) \(\ds \sum_{k \mathop > 0} \floor {\frac {1000} {5^k} }\) De Polignac's Formula
\(\ds \) \(=\) \(\ds \floor {\frac {1000} 5} + \floor {\frac {1000} {25} } + \floor {\frac {1000} {125} } + \floor {\frac {1000} {625} }\)
\(\ds \) \(=\) \(\ds 200 + 40 + 8 + 1\)
\(\ds \) \(=\) \(\ds 249\)

$\blacksquare$