Exponent of primes in factorial n Let $n$ be a natural number and $V2,V3$ and $V5$ denote the exponent of $2,3$ and $5$ in $n!$ respectively. Then it true that $(2^{V2})^2(3^{V3})^2(5^{V5})^2>n!$. I have verified it by calculator. How do I prove it mathematically?

This is not true, for example $59!=2^{54}\cdot 3^{27} \cdot 5^{13}\cdot d$ where $d$ is coprime to $2,3$ and $5$ (see factorize 59!), yet
$$
(2^{54}\cdot 3^{27} \cdot 5^{13})^2 \not > 59!,
$$
see is (2^54 * 3^27 * 5^13)^2 > 59!.
