A box measuring 54 inches long by 36 inches wide by 12

Question

A box measuring 54 inches long by 36 inches wide by 12 inches deep is to be filled entirely with identical cubes. No space is to be left unfilled. What is the smallest number of cubes that can accomplish this objective?

A. 17
B. 18
C. 54
D. 108
E. 864

you need the highest edge of the cube to ensure smallest number of cubes fit in the box.

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Official Answer

D

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Vips0000 · Accepted Answer

least number of cubes will be required when the cubes that could fit in are biggest.
6 is the biggest number that could divide all three, 54, 36 and 12. 
Thus side of cube must be 6, and total number of cubes = 54/6 * 36/6*12/6 = 108

Ans D it is.

PiyushK · Answer

Perfect use of HCF, I solved wrong and got 18. Later realized that 6 is the highest common factor of these numbers.

.... silly conceptual mistake, hope I will not do it on G day.

yezz · Answer

use volumes

volume of box = 54 (2*3^3) * 36 (2^2*3^2) * 12 (2^2*3) = 2^5*3^6

the greatest cub that divide 2^5*3*6 is 2^3*3^3 and the remainder is thus the biggest number of cubes is 2^2*3^3 = 27*4 = 108

PareshGmat · Answer

Find HCF of 54, 36 & 12 = 6

\frac{54 * 36 * 12}{6 * 6 * 6} 
= 108

Answer = D = 108

mvictor · Answer

try to see which kind of cubes can fit it the box, so that no empty space is there:
let's see the common factors of 12, 36, and 54
6, 3, 1
12 can't be, since 54 is not divisible by 12.
9 can't be since 12 is not divisible by 9.

so, to minimize the number of cubes, we need to have cubes with the maximum possible edge - 6. 
12/6 = 2 
36/6 = 6
54/6 = 9
2*6*9 = 108. D.

JeffTargetTestPrep · Answer

The smallest number of identical cubes that can fit into the box without any space left unfilled is one with an edge that is the greatest common factor (GCF) of the three dimensions of the box. Since the dimensions of the box are 54, 36 and 12, their GCF is 6. We should fit 6-inch cubes inside the box, and the number of cubes we can fit is:

(54 x 36 x 12)/(6 x 6 x 6) = 9 x 6 x 2 = 108

Answer: D

akara2500 · Answer

Hi  JeffTargetTestPrep 
Do we need to assume that the edge of the cube will be INTEGER?

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A box measuring 54 inches long by 36 inches wide by 12

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