HMIC 2018 Problem 1

Let $m>1$ be a fixed positive integer. For a nonempty string of base-ten digits $S$, let $c(S)$ be the number of ways to split $S$ into contiguous nonempty strings of digits such that the base-ten number represented by each string is divisible by $m$. These strings are allowed to have leading zeroes. In terms of $m$, what are the possible values that $c(S)$ can take? For example, if $m=2$, then $c(1234)=2$ as the splits 1234 and $12 \mid 34$ are valid, while the other six splits are invalid.