How many cubes of edge 3 cm can be cut from a larger cube of edge 18 cm (no wastage)?

Introduction / Context:
When cutting equal cubes from a larger cube without waste, the count along each edge equals the integer ratio of edge lengths. The total number is the cube of that ratio because packing occurs in three independent directions. This tests spatial counting with simple integer ratios.

Given Data / Assumptions:

• Large cube edge A = 18 cm.
• Small cube edge a = 3 cm.
• Perfect divisibility without kerf loss or waste.

Concept / Approach:
Number per edge = A / a = 18 / 3 = 6. Total cubes = 6^3 because stacking occurs in 3D (length, breadth, height).

Step-by-Step Solution:
Count per edge = 18 / 3 = 6Total = 6 * 6 * 6 = 216

Verification / Alternative check:
Volume ratio: (18^3) / (3^3) = 5832 / 27 = 216, consistent with discrete packing since edges divide exactly.

Why Other Options Are Wrong:
63 and 432 are common miscomputations (e.g., squaring instead of cubing or doubling an intermediate count). “None” is unnecessary because 216 is exact.

Common Pitfalls:
Multiplying only two dimensions (6^2 = 36); forgetting 3D packing; or misreading edge units.

Final Answer:
216