De Polignac's Formula/Examples/7 in 1000

Example of Use of De Polignac's Formula

The prime factor $7$ appears in $1000!$ to the power of $164$.

That is:

$7^{164} \divides 1000!$

but:

$7^{165} \nmid 1000!$

Proof

Let $\mu$ denote the power of $7$ which divides $1000!$

\(\ds \mu\)

\(=\)

\(\ds \sum_{k \mathop > 0} \floor {\frac {1000} {7^k} }\)

De Polignac's Formula

\(\ds \)

\(=\)

\(\ds \floor {\frac {1000} 7} + \floor {\frac {1000} {49} } + \floor {\frac {1000} {343} }\)

\(\ds \)

\(=\)

\(\ds 142 + 20 + 2\)

\(\ds \)

\(=\)

\(\ds 164\)

$\blacksquare$