What is the last nonzero digit to the right of the decimal point in the decimal expansion of $\frac{137}{500}$?
I need to find the decimal expansion of $\frac{137}{500}$ first, so I'll divide 137 by 500 using long division. 500 goes into 137 zero times, so I write a zero in the quotient and a decimal point after it. To continue dividing, I bring down a zero after the decimal point and ask how many times 500 goes into 1370. 500 goes into 1370 two times, with a remainder of 370, so I write a two in the quotient and subtract 500 times 2 from 1370 to get 370. I bring down another zero and ask how many times 500 goes into 3700. 500 goes into 3700 seven times, with a remainder of 200, so I write a seven in the quotient and subtract 500 times 7 from 3700 to get 200. I bring down another zero and ask how many times 500 goes into 2000. 500 goes into 2000 four times, with no remainder, so I write a four in the quotient and stop dividing, since I have no more digits to bring down. The decimal expansion of $\frac{137}{500}$ is 0.274, so the last nonzero digit to the right of the decimal point is 4. 