I would first determine the least common multiple (LCM) of R and P, which is almost entry level algorithmic. Then divide S by the LCM, and multiply the LCM by the integer part of the result of that division.

That's effectively, exactly what I was doing, but in my sleep-deprived state, I was convinced that I could avoid the iteration and calculate the result directly, despite that I couldn't see how. Hence my question. Thank you.