A cuboidal block of 6 cm x 9 cm x 12 cm is cut up into an exact number of equal cubes. The least possible number of cubes will be

### Concept & HCF for Minimum Divisions To cut a block into the least possible number of identical cubes, the cubes must be as large as possible. The maximum edge length of such a cube is the Highest Common Factor (HCF) of the block's dimensions. $$ \text{Maximum Edge Length} = \text{HCF}(L, B, H) $$ ### Step-by-Step Solution * Dimensions of the cuboidal block: Length = $12\text{ cm}$, Breadth = $9\text{ cm}$, Height = $6\text{ cm}$. * Find the HCF of 12, 9, and 6: The largest number that divides all three evenly is 3. So, the maximum edge of the cube is $3\text{ cm}$. * Find the volume of one such cube: $3 \times 3 \times 3 = 27\text{ cm}^3$. * Find the volume of the block: $6 \times 9 \times 12 = 648\text{ cm}^3$. * Divide the block's volume by the cube's volume: $\frac{648}{27} = 24$ cubes. ### Exam Strategy & Shortcut Instead of multiplying volumes, divide each dimension by the HCF (3) and multiply the results: $\left(\frac{6}{3}\right) \times \left(\frac{9}{3}\right) \times \left(\frac{12}{3}\right) = 2 \times 3 \times 4 = 24$. ### Common Pitfall Taking the LCM instead of the HCF, which is the strategy used for building a large solid from smaller blocks (like the next question type), rather than cutting a solid down. ### Final Answer Therefore, the correct answer is **24**.