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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \floor {\frac {1000} 5} + \floor {\frac {1000} {25} } + \floor {\frac {1000} {125} } + \floor {\frac {1000} {625} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 200 + 40 + 8 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 249\) |
$\blacksquare$