Note that $S(889)=25$ and $S(790321)=22$. We now have to consider the following sequence: $a_1=889, a_n = a_{n-1}*10^8+889$. It will hold that $S(a_n)-S(a_n^2)=25*n - 22*n=3n$. So we take $n=\lceil m/3\rceil +1$ and we are done.

enrique17 wrote: Note that $S(889)=25$ and $S(790321)=22$. We now have to consider the following sequence: $a_1=889, a_n = a_{n-1}*10^8+889$. It will hold that $S(a_n)-S(a_n^2)=25*n - 22*n=3n$. So we take $n=\lceil m/3\rceil +1$ and we are done. $S(a_n^2)\ne 22n$ because of the term $2a_{n-1}*10^8*889$.