Multiples of repunits as sum of powers of ten

The sequence Pk,n = 1 + 10k + 102k + ⋯ + 10(n−1)k can be used to generate infinitely many Smith numbers with the help of a set of suitable multipliers. We prove the existence of such a set, consisting of constant multiples of repunits, that generalizes to any value of k ⩾ 9. This fact complements the earlier results which have been established for k ⩽ 9.