If the students of a class can be grouped exactly into 6 or 8 or 10, then the minimum number of student 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 calculate the Least Common Multiple (LCM) of these three numbers.

### Step-by-Step Solution:

1. **Identify the numbers**: We need to find the LCM of 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 power of each prime**:
- The prime factors we have 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 = \(2^3 \times 3^1 \times 5^1\)
- Calculate each part:
- \(2^3 = 8\)
- \(3^1 = 3\)
- \(5^1 = 5\)
- Now multiply these together:
- \(LCM = 8 \times 3 = 24\)
- \(24 \times 5 = 120\)

5. **Conclusion**:
- The minimum number of students in the class must be 120, as it can be grouped into 6, 8, or 10 students.

### Final Answer:
The minimum number of students in the class is **120**.

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