Suppose (p,q) = 1, (p,10)=1 and (q,10)=1. if 1/p has period r and 1/q has period s find the period of 1/pq.
Can anyone help - I've got the start but seem to have become stuck. Any help is much appreciated.
If r is the period of 1/p then r is the smallest integer such that p divides $\displaystyle 10^r-1$. (Examples: $\displaystyle 1/11 = 0.\overline{09}$ has period 2, and 11 divides $\displaystyle 99 = 10^2-1$; $\displaystyle 1/37 = 0.\overline{027}$, and 37 divides $\displaystyle 999=10^3-1$.)
If 1/p has period r and 1/q has period s, and (p,q) = 1, then the smallest number of the form $\displaystyle 10^t-1$ that is a multiple of pq ought to be given by $\displaystyle t=\text{lcm}(r,s)$.
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.