The size of a wooden block is 5 x 10 x 20 cm. How many such blocks will be required to construct a solid wooden cube of minimum size?

### Concept & LCM for Constructing Solids To construct the smallest possible solid cube from identical rectangular blocks, the side length of the cube must be the Least Common Multiple (LCM) of the block's dimensions. $$ \text{Minimum Edge Length} = \text{LCM}(L, B, H) $$ ### Step-by-Step Solution * Dimensions of the given wooden block: $5\text{ cm}$, $10\text{ cm}$, and $20\text{ cm}$. * Find the LCM of 5, 10, and 20: The smallest number divisible by all three is 20. So, the edge of the minimum solid cube is $20\text{ cm}$. * Calculate the volume of this constructed cube: $20 \times 20 \times 20 = 8000\text{ cm}^3$. * Calculate the volume of one wooden block: $5 \times 10 \times 20 = 1000\text{ cm}^3$. * Divide the total volume by the block volume to find the count: $\frac{8000}{1000} = 8$ blocks. ### Exam Strategy & Shortcut Find how many blocks fit along each axis of the constructed $20\text{ cm}$ cube: $\frac{20}{5} = 4$ blocks, $\frac{20}{10} = 2$ blocks, $\frac{20}{20} = 1$ block. Total blocks = $4 \times 2 \times 1 = 8$. ### Common Pitfall Finding the HCF instead of LCM, which would mistakenly apply the logic of cutting a block rather than building up to a cube. ### Final Answer Therefore, the correct answer is **8**.