A cuboidal block with dimensions 6 cm × 9 cm × 12 cm is cut up into an exact number of equal smaller cubes. What is the least possible number of equal cubes obtained?

Introduction / Context:
This problem involves partitioning a cuboid into smaller cubes of equal size. To obtain the least number of equal cubes, we must use the largest possible edge length for the small cubes that still divides each of the three dimensions of the original block exactly. This approach relies on the concept of the greatest common divisor (GCD) for the dimensions and basic volume reasoning.

Given Data / Assumptions:

• Dimensions of cuboidal block: 6 cm × 9 cm × 12 cm.
• The block is cut into identical cubes, each with side length s cm.
• The number of cubes must be an integer and as small as possible.

Concept / Approach:
For cubes to fit exactly, the side length s must divide each dimension 6, 9 and 12 without remainder. The largest such s is the greatest common divisor of these three numbers. Once s is known, the number of cubes equals the ratio of the volume of the cuboid to the volume of a cube:
Number of cubes = (6 * 9 * 12) / s^3

Step-by-Step Solution:
Compute the greatest common divisor of 6, 9 and 12. GCD(6, 9) = 3, GCD(3, 12) = 3, so s = 3 cm. Thus, each small cube has side 3 cm. Volume of cuboid = 6 * 9 * 12 cubic centimetres. Volume of cuboid = 648 cm^3. Volume of one cube = 3^3 = 27 cm^3. Number of cubes = 648 / 27 = 24.

Verification / Alternative check:
We can verify by counting cubes along each dimension. Along length 12 cm, we fit 12 / 3 = 4 cubes. Along breadth 9 cm, we fit 9 / 3 = 3 cubes. Along height 6 cm, we fit 6 / 3 = 2 cubes. Therefore, the total number of cubes is 4 * 3 * 2 = 24, matching the volume-based calculation.

Why Other Options Are Wrong:
Values such as 6, 9, 18 and 30 would correspond to using a smaller cube size or miscounting the cubes along one or more dimensions. For example, 18 would require incorrect division along at least one dimension. Because 3 cm is the largest edge length that divides all three dimensions exactly, any smaller cube would give more than 24 pieces, not fewer.

Common Pitfalls:
A typical mistake is to pick the smallest dimension (6 cm) as the cube side length without checking divisibility with 9 and 12. Others might use the least common multiple instead of the greatest common divisor by mistake. Students also sometimes forget that the number of cubes is a three-dimensional count (product along length, breadth and height) rather than just dividing by one dimension.

Final Answer:
The least possible number of equal cubes obtained is 24.