imhimanshu wrote:
For a display, identical cubic boxes are stacked in square layers. Each layer consists of cubic boxes arranged in rows that form a square, and each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it. If the bottom of the layer has 81 boxes and the top of the layer has only 1 box, how many boxes are in display?
A. 236
B. 260
C. 269
D. 276
E. 285
We see that the bottommost layer has 9 x 9 = 81 boxes and the topmost layer is 1 x 1 = 1 box. Therefore, each layer must be a perfect square number of boxes from 1 to 81, inclusive. That is, if we count the layers from top to bottom, they have 1, 4, 9, 16, 25, 36, 49, 64, and 81 boxes. Therefore, the total number of boxes is:
1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81
Instead of adding the actual numbers, let’s add their units digits:
1 + 4 + 9 + 6 + 5 + 6 + 9 + 4 + 1 = 5 + 15 + 5 + 15 + 5 = 45
We see the sum must have a units digit of 5; therefore, choice E is the correct answer.
Answer: E
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