Parikshit makes a cuboid of plasticine of sides `5 cm, 2 cm, 5 cm`. How many such cuboids will he need to form a cube?

Options

A. `22`
B. `21`
C. `23`
D. `20`

Explanation

To solve the problem of how many cuboids Parikshit needs to form a cube, we will follow these steps:

### Step 1: Calculate the Volume of the Cuboid
The volume \( V \) of a cuboid is calculated using the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
Given dimensions of the cuboid are:
- Length = 5 cm
- Breadth = 2 cm
- Height = 5 cm

Now, substituting the values into the formula:
\[ V = 5 \, \text{cm} \times 2 \, \text{cm} \times 5 \, \text{cm} = 50 \, \text{cm}^3 \]

### Step 2: Determine the Side Length of the Cube
To form a cube, we need to determine the side length of the cube. The side length must be a common multiple of the dimensions of the cuboid. The dimensions are 5 cm, 2 cm, and 5 cm.

To find the least common multiple (LCM) of these dimensions:
- The prime factorization of 5 is \( 5^1 \).
- The prime factorization of 2 is \( 2^1 \).

The LCM is calculated by taking the highest power of each prime factor:
\[ \text{LCM} = 2^1 \times 5^1 = 10 \, \text{cm} \]

### Step 3: Calculate the Volume of the Cube
Now, we can calculate the volume of the cube using the formula:
\[ V = \text{side}^3 \]
Substituting the side length:
\[ V = 10 \, \text{cm} \times 10 \, \text{cm} \times 10 \, \text{cm} = 1000 \, \text{cm}^3 \]

### Step 4: Calculate the Number of Cuboids Needed
To find out how many cuboids are needed to form the cube, we divide the volume of the cube by the volume of one cuboid:
\[ \text{Number of cuboids} = \frac{\text{Volume of the cube}}{\text{Volume of the cuboid}} = \frac{1000 \, \text{cm}^3}{50 \, \text{cm}^3} = 20 \]

### Conclusion
Parikshit will need **20 cuboids** to form a cube.

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