A solid rectangular block has dimensions 6 cm by 12 cm by 15 cm. It is to be cut exactly into a certain number of equal small cubes, all of the same size, with no waste of material. What is the smallest possible number of such cubes?

Introduction / Context:
This is a standard problem on cutting a rectangular solid into the maximum size identical cubes while using the entire volume. The key idea is that the edge of each small cube must be a common measure of all three dimensions of the rectangular block, so that an integer number of cubes fits exactly along each side. The smallest number of cubes will occur when each cube has the largest possible edge length, which is equal to the greatest common divisor of the three dimensions.

Given Data / Assumptions:

• Dimensions of the rectangular block are 6 cm, 12 cm and 15 cm.
• The block is solid and cut completely into equal cubes.
• There is no wastage of material.
• We seek the least possible number of cubes.

Concept / Approach:
Let the side of each small cube be s cm. For the cutting to be exact, s must divide each of 6, 12 and 15 exactly. Therefore s must be a common divisor of the three dimensions. To minimize the number of cubes, we want to maximize s. Hence s equals the greatest common divisor (gcd) of 6, 12 and 15. Once s is found, the number of cubes N is:
N = (6 * 12 * 15) / s^3

Step-by-Step Solution:
Step 1: Compute gcd of 6, 12 and 15. gcd(6, 12) = 6. gcd(6, 15) = 3. So overall gcd is 3. Step 2: Take cube side s = 3 cm. Step 3: Count how many cubes along each dimension. Along 6 cm side: 6 / 3 = 2 cubes. Along 12 cm side: 12 / 3 = 4 cubes. Along 15 cm side: 15 / 3 = 5 cubes. Step 4: Total number of cubes. N = 2 * 4 * 5 = 40. Thus, the least possible number of equal cubes is 40.

Verification / Alternative check:
Using the volume method: Volume of block = 6 * 12 * 15 = 1080 cubic centimetres. Volume of each cube with side 3 cm = 3^3 = 27 cubic centimetres. Number of cubes N = 1080 / 27 = 40. The result matches the earlier computation, confirming correctness.

Why Other Options Are Wrong:
Option A (30) suggests a smaller number of cubes, which would require a larger cube size than 3 cm that still divides all dimensions, but no such divisor exists. Option C (10) and Option D (20) correspond to still larger cube sizes and are not compatible with all three dimensions. Option E (60) corresponds to a smaller cube size, which would increase, not decrease, the number of cubes.

Common Pitfalls:
Some students mistakenly choose the smallest dimension as the cube edge, or use the least common multiple instead of the greatest common divisor. Others forget that the cube size must divide all three dimensions exactly. Always check divisibility and use the gcd to achieve the largest cube size and smallest count.

Final Answer:
The smallest possible number of equal cubes is 40.