Number Theory

In order to solve this problem, we can use a very simple trick. 

First, let's set $x = 0.\overline{2123}$

If we have this for the value of x, we get $10000 x = 2123.\overline{2123}$

Now, we subtract x from 10000x. We get

$10000x-x= 2123.\overline{2123} -\overline{2123}\\ 9999x=2123$

This means we get

$x = \frac{2123}{9999}$

Simplifying, we get

$x = \frac{193}{909}$

So our final answer is $\frac{193}{909}$

Thanks! :)

The answer is 17/210.