# Question #d237f

Jan 26, 2015

I'd say 99.

In fact, a periodic number like $0. a b c a b c a b c \ldots$ is given by the fraction $\frac{a b c}{999}$, and in general such periodic numbers are given by the fraction with the repeating block at the numerator and as many nines at the denominator as the lenght of the repeating block.

So, for example, 0.125125125.. will be given by $\frac{125}{999}$, 0.3434... will be given by $\frac{34}{99}$, and so on.

So, any 2-digits number $x y$, divided by 99, will give the periodic number $0. x y x y x y \ldots$

Note that this works in some "strange" cases:

1. by choosing $x y = 0$, one has that $\frac{0}{99} = 0$, which you can read as 0.000....
2. for any number from 1 to 9, this formula works if you remember that you are thinking of them as a 2-digits number, so 1 is 01, 2 is 02, and so on. In fact you have that, for example, $\frac{5}{99} = 0.0505 \ldots$
3. Probably the trickiest point is that, even though you asked $x y < a b$, this actually works also for $x y = 99$. In fact, $\frac{99}{99}$ is of course 1, but $1 = 0.99 \ldots$