What is the $2007^{\text{th}}$ digit to the right of the decimal point in the decimal expansion of $\frac{1}{7}$?
I know that the decimal expansion of $\frac{1}{7}$ is $0.\overline{142857}$, which means that it repeats the six digits $142857$ infinitely. So, to find the $2007^{\text{th}}$ digit, I need to figure out which cycle of six digits it belongs to, and then which position within that cycle. I can do this by dividing $2007$ by $6$ and looking at the quotient and the remainder. If I divide $2007$ by $6$, I get $334$ with a remainder of $3$. This means that the $2007^{\text{th}}$ digit is in the $335^{\text{th}}$ cycle of six digits, and it is the third digit in that cycle. The third digit in each cycle is $2$, so that is the answer. 