What is the product of the numerator and the denominator when $0.\overline{009}$ is expressed as a fraction in lowest terms?
I know that to express a repeating decimal as a fraction, I can use a clever trick to eliminate the repeating part. In this case, the repeating part is 009, which has a period of 3 digits. So I can multiply both sides of the equation $0.\overline{009} = x$ by 1000, which shifts the decimal point 3 places to the right. This gives me $9.\overline{009} = 1000x$. Now, if I subtract the original equation from this one, I get $9 = 999x$. This simplifies to $x = \frac{9}{999}$. I can reduce this fraction by dividing both the numerator and the denominator by 9, which gives me $x = \frac{1}{111}$. Therefore, the product of the numerator and the denominator is 1 * 111, which is 111. 