De Polignac's Formula/Examples/13 in 1000

Example of Use of De Polignac's Formula

The prime factor $13$ appears in $1000!$ to the power of $81$.

That is:

$13^{81} \divides 1000!$

but:

$13^{82} \nmid 1000!$

Proof

Let $\mu$ denote the power of $13$ which divides $1000!$

\(\ds \mu\)

\(=\)

\(\ds \sum_{k \mathop > 0} \floor {\frac {1000} {13^k} }\)

De Polignac's Formula

\(\ds \)

\(=\)

\(\ds \floor {\frac {1000} {13} } + \floor {\frac {1000} {169} }\)

\(\ds \)

\(=\)

\(\ds 76 + 5\)

\(\ds \)

\(=\)

\(\ds 81\)

$\blacksquare$