In a seminar the number of participants in Mathematics, Physics and Biology are 192, 240 and 168. Find the minimum number of rooms required, if in each room same number of participants is to be seated and all of them being in the same subject.

Options

A. 20
B. 25
C. 28
D. 30

Answer

To find the minimum number of rooms required for the seminar participants in Mathematics, Physics, and Biology, we need to follow these steps:

### Step 1: Identify the number of participants in each subject.
- Mathematics: 192 participants
- Physics: 240 participants
- Biology: 168 participants

### Step 2: Find the Highest Common Factor (HCF) of the three numbers.
To do this, we will factor each number into its prime factors.

1. **Factor 192:**
- 192 = 2 × 96
- 96 = 2 × 48
- 48 = 2 × 24
- 24 = 2 × 12
- 12 = 2 × 6
- 6 = 2 × 3
- Therefore, 192 = 2^6 × 3^1

2. **Factor 240:**
- 240 = 2 × 120
- 120 = 2 × 60
- 60 = 2 × 30
- 30 = 2 × 15
- 15 = 3 × 5
- Therefore, 240 = 2^4 × 3^1 × 5^1

3. **Factor 168:**
- 168 = 2 × 84
- 84 = 2 × 42
- 42 = 2 × 21
- 21 = 3 × 7
- Therefore, 168 = 2^3 × 3^1 × 7^1

### Step 3: Determine the HCF.
To find the HCF, we take the lowest power of each prime factor present in all three factorizations:

- For the prime factor 2: The minimum power is 2^3 (from 168).
- For the prime factor 3: The minimum power is 3^1 (common in all).
- The prime factors 5 and 7 are not common in all three numbers.

Thus, the HCF = 2^3 × 3^1 = 8 × 3 = 24.

### Step 4: Calculate the number of rooms required for each subject.
Now, we will divide the total number of participants in each subject by the HCF (24) to find the number of rooms needed.

1. **Mathematics:**
- Number of rooms = 192 / 24 = 8 rooms

2. **Physics:**
- Number of rooms = 240 / 24 = 10 rooms

3. **Biology:**
- Number of rooms = 168 / 24 = 7 rooms

### Step 5: Calculate the total number of rooms.
Total rooms required = 8 (Mathematics) + 10 (Physics) + 7 (Biology) = 25 rooms.

### Final Answer:
The minimum number of rooms required is **25**.

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