I am conducting a little research on checking if a number, written in positional numeral system is irrational.

Let $h^p_n$ be the most right non-zero digit of number $n!$ written in numeral system with base $p$. For example, in system with base $10$, $8!=40320\Rightarrow h_8^{10}=2.$ The question is following: for which $p$ number $H^p=0,h^p_1h^p_2\ldots h^p_n\ldots$ is irrational?

Obviously, $H^2$ is rational. I also know that $H^{10}$ is irrational, but have no idea on how to prove it and would really appreciate any kind of hint.