Correct Option (3)
Both Statement I and Statement II are correct.
Statement I: The minimum number of cuts is 9.
When a cube is cut into smaller pieces, if it is cut 'x' times along its length, 'y' times along its width, and 'z' times along its height, the total number of smaller pieces formed is given by the product (x + 1)(y + 1)(z + 1). Given that the cube is cut into 60 smaller pieces, we have:
(x + 1)(y + 1)(z + 1) = 60
To minimize the total number of cuts (x + y + z), the factors (x + 1), (y + 1), and (z + 1) should be as close to each other as possible. The prime factorization of 60 is 2 × 2 × 3 × 5. The combination of three factors closest to each other is 3, 4, and 5.
- Let x + 1 = 3 ⇒ x = 2
- Let y + 1 = 4 ⇒ y = 3
- Let z + 1 = 5 ⇒ z = 4
The minimum number of cuts required is x + y + z = 2 + 3 + 4 = 9. Thus, Statement I is correct.
Statement II: The number of smaller pieces which are not painted on any face is 6.
The smaller pieces that are not painted on any face are the internal pieces, which do not form part of the outermost layer of the original cube. If the cube is divided into (x + 1), (y + 1), and (z + 1) segments along its dimensions, the number of unpainted pieces is given by (x + 1 - 2)(y + 1 - 2)(z + 1 - 2).
Using the values derived from Statement I, where (x + 1) = 3, (y + 1) = 4, and (z + 1) = 5:
Number of unpainted pieces = (3 - 2)(4 - 2)(5 - 2) = 1 × 2 × 3 = 6.
Thus, Statement II is correct.
Incorrect Options:
Options 1, 2, and 4 are incorrect because the analysis confirms that both Statement I and Statement II are factually accurate, leading to the conclusion that option 3 is the correct choice.
