If the students of a class can be grouped exactly into 6 or 8 or 10, then the minimum number of students in the class must be

Options

A. 60
B. 120
C. 180
D. 240

Answer

To find the minimum number of students in the class that can be grouped exactly into 6, 8, or 10, we need to determine the least common multiple (LCM) of these numbers.

### Step-by-Step Solution:

1. **Identify the numbers**: The numbers we need to find the LCM for are 6, 8, and 10.

2. **Prime Factorization**:
- For 6: \(6 = 2^1 \times 3^1\)
- For 8: \(8 = 2^3\)
- For 10: \(10 = 2^1 \times 5^1\)

3. **Determine the highest powers of each prime**:
- The prime factors involved are 2, 3, and 5.
- The highest power of 2 is \(2^3\) (from 8).
- The highest power of 3 is \(3^1\) (from 6).
- The highest power of 5 is \(5^1\) (from 10).

4. **Calculate the LCM**:
- LCM = \(2^3 \times 3^1 \times 5^1\)
- LCM = \(8 \times 3 \times 5\)

5. **Perform the multiplication**:
- First, calculate \(8 \times 3 = 24\).
- Then, calculate \(24 \times 5 = 120\).

6. **Conclusion**: The minimum number of students in the class must be **120**.