For a prime and a given integer let denote the exponent of in the prime factorisation of . Given and a set of primes, show that there are infinitely many positive integers such that for all .
I don't know if this helps at all, but apparently there is a formula due to Legendre saying that $\displaystyle
\nu_p(n!) = \frac{n-s_p(n)}{p-1}$, where $\displaystyle s_p(n)$ is the sum of the digits of n in base p.
Reference: http://www.math.tulane.edu/~vhm/web_html/k-central.pdf (pdf file).