  Cobham's theorem asserts that if a sequence is automatic with respect to two
multiplicatively independent bases, then it is ultimately periodic. We prove a
stronger density version of the result: if two sequences which are automatic
with respect to two multiplicatively independent bases coincide on a set of
density one, then they also coincide on a set of density one with a periodic
sequence. We apply the result to a problem of Deshouillers and Ruzsa concerning
the least nonzero digit of $n!$ in base $12$.
