The third-named author recently proved
[Israel J. of Math. 258 (2023), 475–502] that there are infinitely many
collisions of the base-2 and base-3 sum-of-digits functions.
In other words, the equation

$$
s_2(n)=s_3(n)
$$

admits infinitely many solutions in natural numbers. We refine this
result and prove that every integer
\( a \in \{1,2,\ldots,11\} \)
appears as the last nonzero digit of \( n! \) in base 12 infinitely often.