Competition problem — Number theory

Library / /240 of 520

Number theory Difficulty 6.0 National olympiad Prove it

Property 5 Let $p$ be a prime, $n \in \mathbf{N}^{*}$ , then
$v_{p}(n!)=\left[\frac{n}{p}\right]+\left[\frac{n}{p^{2}}\right]+\cdots$

Solution

None

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.

Note: The provided instruction is a meta-instruction and not part of the text to be translated. Since the text to be translated is "None", the translation is also "None". Here is the formatted output as requested:

None

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.