If r is the period of 1/p then r is the smallest integer such that p divides . (Examples: has period 2, and 11 divides ; , and 37 divides .)

If 1/p has period r and 1/q has period s, and (p,q) = 1, then the smallest number of the form that is a multiple of pq ought to be given by .

So the answer should be that the period of 1/(pq) is lcm(r,s). I'll leave you to fill in the details of that argument.