AMC8 1991

AMC8 1991 · Q24

AMC8 1991 · Q24. It mainly tests 3D geometry (volume), Divisibility & factors.

A cube of edge 3 cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N =$

一个边长3 cm的立方体被切成$N$个较小的立方体，不全相同大小。如果每个小立方体的边长是整厘米，则$N =$

(A) 4 4

(B) 8 8

(D) 16 16

(E) 20 20

Answer

Correct choice: (E)

正确答案：（E）

Solution

Since the edge of each smaller cube must be a whole number, the smaller cubes must be $1\times1\times1$ or $2\times2\times2$ cubes. There can be only one smaller cube of edge 2, so the rest of the smaller cubes have edge 1. Since the volume of the original cube was $3 \times 3 \times 3 = 27$ cubic cm, and the volume of the cube of edge 2 is $2 \times 2 \times 2 = 8$ cubic cm, then there must be $27 - 8 = 19$ cubes of edge 1 (volume = 1 cubic cm). There are a total of 20 cubes so $N = 20$.

由于每个小立方体的边长必须是整数，所以小立方体必须是$1\times1\times1$或$2\times2\times2$。边长2的小立方体只能有一个，其余是边长1的。原立方体体积$3 \times 3 \times 3 = 27$立方厘米，边长2的立方体体积$2 \times 2 \times 2 = 8$立方厘米，因此边长1的立方体有$27 - 8 = 19$个（体积1立方厘米）。总共有20个立方体，所以$N = 20$。

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