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Question The length, width, and height of a box are 12 cm, 18 cm, and 36 cm, respectively. Which of the following-sized cubes can be packed in this box without leaving gaps?
(a) 9 cm
(b) 6 cm
(c) 4 cm
(d) 3 cm
(e) 2 cm
Here, dimensions of box: Length = 12 cm, Width = 18 cm, Height = 36 cm The size of the largest cube that can exactly fit (without gaps) That means the side of the cube must exactly divide all three dimensions of the box. So, we need to find the HCF (Highest Common Factor) of 12, 18, and 36. Prime factorisation 12 = 2 × 2 × 3, 18 = 2 × 3 × 3, and 36 = 2 × 2 × 3 × 3 Common factors = 2 × 3 = 6 HCF = 6 cm The cube must have a side length equal to a factor of the HCF. From the options: (a) 9 cm ✗ (9 doesn’t divide 12 evenly) (b) 6 cm ✓ (divides 12, 18, and 36 exactly) (c) 4 cm ✗ (doesn’t divide 18 evenly) (d) 3 cm ✓ (also divides all) (e) 2 cm ✓ (also divides all) Hence, the largest possible cube that fits without gaps is (b) 6 cm.