Automaticity of the sequence of the last nonzero digits of $n!$ in a fixed base

TL;DR

This paper investigates the automaticity of the sequence of last nonzero digits of factorials in various bases, providing a new proof and a complete characterization of when this sequence is automatic.

Contribution

It offers an alternative proof to previous results and generalizes the problem to characterize all bases for which the sequence is automatic.

Findings

The sequence is not automatic in base 12.

Provides a complete characterization of bases with automatic last nonzero digit sequences.

Generalizes previous results to a broader class of bases.

Abstract

In 2011 Deshouillers and Ruzsa tried to argument that the sequence of the last nonzero digit of $n!$ in base 12 is not automatic. This statement was proved few years later by Deshoulliers. In this paper we provide alternate proof that lets us generalize the problem and give an exact characterization in which bases the sequence of the last nonzero digits of $n!$ is automatic.