This problem permits a really short recursive solution.
Generally I prefer non-recursive approaches, as recursion tends to be slow (even with memo) and recursion depth limits may apply. However recursive solutions give more concise and elegant code.

Python3:

from functools import cache
def stoneDivision(n, s):
    @cache
    def val(x):
        divs = [i for i in s if x%i == 0 and i!=x]
        return max(1 + val(i) * x//i for i in divs) if divs else 0
    return val(n)