Question

A cuboidal block, 12 cm by 24 cm by 30 cm, is cut up into an exact number of identical cubes. The least possible number of such cubes is:

• A. 120
• B. 100
• C. 110
• D. 140

Answer 1

Answer:

A. 120

The least possible number of cubes is 120.

To minimize the number of cubes, each cube must be as large as possible. This means the side length of the cube must be the Highest Common Factor (HCF) of the cuboid's dimensions (12 cm, 24 cm, and 30 cm).

Step 1: Find the maximum side length of a cube (HCF)

• Factors of 12: $2 \times 2 \times 3$

• Factors of 24: $2 \times 2 \times 2 \times 3$

• Factors of 30: $2 \times 3 \times 5$

The common prime factors are 2 and 3.
$\text{HCF} = 2 \times 3 = \mathbf{6\text{ cm}}$

So, the maximum size of each identical cube is 6 cm $\times$ 6 cm $\times$ 6 cm.

Step 2: Calculate the number of cubes
Divide the total volume of the cuboid by the volume of one cube:
$\text{Number of cubes} = \frac{\text{Volume of Cuboid}}{\text{Volume of Cube}}$

$\text{Number of cubes} = \frac{12 \times 24 \times 30}{6 \times 6 \times 6}$

$\text{Number of cubes} = 2 \times 4 \times 5 = \mathbf{120}$