How many rectangular pieces of size 5 cm × 5 cm × 10 cm can be cut from a larger cake of size 5 cm × 30 cm × 30 cm (no waste by thickness mismatch)?

Introduction / Context:
Cutting identical rectangular blocks from a larger rectangular block without waste is a packing-by-integers problem. The count equals the product of how many blocks fit along each dimension, using an orientation that matches the dimensions as integers.

Given Data / Assumptions:

• Big cake: 5 × 30 × 30 (cm).
• Small piece: 5 × 5 × 10 (cm).
• Reorientation of the small piece dimensions is allowed.

Concept / Approach:
Choose orientation so each small dimension divides the corresponding big dimension. A natural fit is: along thickness 5 → 5, along one length 5 → 30, along the other 10 → 30.

Step-by-Step Solution:
Along 5: ⌊5/5⌋ = 1Along 30 with 5: ⌊30/5⌋ = 6Along 30 with 10: ⌊30/10⌋ = 3Total pieces = 1 * 6 * 3 = 18

Verification / Alternative check:
Other orientations (e.g., 10 along thickness) would waste the 5-thickness dimension; the chosen orientation achieves exact fits on all sides.

Why Other Options Are Wrong:
10 and 15 undervalue the packing; 30 would require fitting 6 along both 30-dimensions and multiple along thickness, which the 10-cm side cannot support.

Common Pitfalls:
Not considering rotation of the small block; forgetting to take floors of the ratios for each dimension.

Final Answer:
18