# Math Help - Period of fraction - Number Theory

1. ## Period of fraction - Number Theory

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.

2. Originally Posted by jp3105
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 $10^r-1$. (Examples: $1/11 = 0.\overline{09}$ has period 2, and 11 divides $99 = 10^2-1$; $1/37 = 0.\overline{027}$, and 37 divides $999=10^3-1$.)

If 1/p has period r and 1/q has period s, and (p,q) = 1, then the smallest number of the form $10^t-1$ that is a multiple of pq ought to be given by $t=\text{lcm}(r,s)$.

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.