jmc

If we define the variable $s$ to be $0.\overline{123}$, then multiplying both sides of $s=0.\overline{123}$ by 1000 gives us $$1000s=123.\overline{123}.$$ Subtracting $s$ from $1000s$ and $0.\overline{123}$ from $123.\overline{123}$ tells us that $$999s=123$$ and thus $$s=\frac{123}{999}.$$ We can now calculate that our final answer is $$\frac{123}{999}\cdot 9=\frac{123}{999÷9}=\frac{123÷3}{111÷3}=\boxed{\frac{41}{37}}.$$