Example 8. A merchant has a 40-pound weight, which broke into 4 pieces after falling to the ground. Later, it was found that the weight of each piece is an integer number of pounds, and these 4 pieces can be used to weigh any integer weight from 1 to 40 pounds. What are the weights of these 4 pieces of the weight?
To make two weights A and B measure the most weight, you can choose A=1, B=3, with which you can measure 1, 2, 3, 4 pounds of objects. If you choose a third weight C=2×4+1=9, then you can use A, B, C to measure objects from 1 to 13 (4+9=13) pounds. Finally, if you choose a fourth weight D=2×13+1=27, then you can use A, B, C, D to measure objects from 1 to 40 (13+27=40) pounds. Therefore, the four weights are 1, 3, 9, 27 pounds.

Note: There are two scenarios for using weights to measure objects. One is where the weights can only be placed on the weight pan (only addition is allowed), and the reasonable weights are \(2^{0}, 2^{1}, 2^{2}, 2^{3}, \cdots\); the other is where the weights can be placed in either pan (both addition and subtraction are allowed), and the reasonable weights are \(3^{0}, 3^{1}, 3^{2}, 3^{3}, \cdots\).