Find the $291^{\text{st}}$ digit past the decimal point in the expansion of $\dfrac{1}{37}$.
Then I divide 37 into 1 and get 0 with a remainder of 1. I bring down a zero and divide 37 into 10 and get 0 with a remainder of 10. I bring down another zero and divide 37 into 100 and get 2 with a remainder of 26. I bring down another zero and divide 37 into 260 and get 7 with a remainder of 1. I write the quotient as 0.027 and put a bar over the 027 to indicate that it repeats indefinitely. So the decimal expansion of $\dfrac{1}{37}$ is 0.027027027... One way to do that is to divide 291 by 3, since the repeating part has a length of 3. I get 291 divided by 3 is 97 with a remainder of 0. This means that the $291^{\text{st}}$ digit is the last digit of the 97th repetition of 027. The last digit of 027 is 7, so that is the answer. 