CDS 2026 (I) Elementary Mathematics, question 69

Topic: Mensuration

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Q69

The volume of a cube is $8\text{ cm}^{3}$ and is equal to the volume of a cuboid of length $x$ , breadth $y$ and height $z$ , where $x,y$ and $z$ are natural numbers $(x>y>z)$ . Let $n$ be the number of such cuboids with different dimensions. What is the value of $n$ ?

(a) $1$ Correct answer
(b) $2$
(c) $3$
(d) More than 3

Solution.

Answer: (a) $1$

Since $xyz = 8$ and $x > y > z$ are positive integers, z cannot be at least 2: that would force $xyz \geq 4 \times 3 \times 2 = 24$ .
Hence $z = 1$ .
Now $xy = 8$ with $x > y > 1$ .
Only $(x,y) = (4,2)$ works; (8, 1) repeats 1 and violates the strict inequalities.