There is a cube of volume 216 cm³. It is to be moulded into a cuboid having one edge equal to 6 cm. The number of ways that it can be done so that the edges have different integral values is

### Concept & Volume Dimensions The volume of a cuboid is the product of its three dimensions (length, breadth, and height). When a shape is remoulded, its volume remains constant. $$V = l \times b \times h$$ ### Step-by-Step Solution 1. **Given:** Volume of the original cube = $216 \text{ cm}^3$. This means the new cuboid also has a volume of $216 \text{ cm}^3$. 2. One edge of the cuboid is given as $6 \text{ cm}$. Let the other two edges be $l$ and $b$. 3. Thus, $l \times b \times 6 = 216$. 4. Dividing by 6, we get $l \times b = 36$. 5. We need to find pairs of integers $(l, b)$ whose product is 36. The possible factor pairs of 36 are: - $(1, 36)$ - $(2, 18)$ - $(3, 12)$ - $(4, 9)$ - $(6, 6)$ 6. The question specifies that the edges must have **different integral values**. The three edges are $l$, $b$, and $6$. 7. This means neither $l$ nor $b$ can be $6$, which rules out the $(6, 6)$ pair (where edges would be $6, 6, 6$). 8. Checking the remaining pairs against the third edge ($6$): - $1, 36, 6$ (all different) - Valid - $2, 18, 6$ (all different) - Valid - $3, 12, 6$ (all different) - Valid - $4, 9, 6$ (all different) - Valid 9. There are 4 valid ways to form the cuboid. ### Exam Strategy & Shortcut To quickly solve constraint-based factor problems, instantly list out all factor pairs of the remaining product area ($36$). Then apply constraints: eliminate pairs with duplicates (like $6 \times 6$) or pairs matching the constant given edge. Count what remains. ### Common Pitfall A common mistake is failing to check the generated pairs against the fixed edge of $6 \text{ cm}$ or forgetting that $(6,6)$ creates a shape with identical edges, violating the "different integral values" condition. ### Final Answer Therefore, the correct answer is **4**.