6, 8, 12 or 16 rows of the ninth class students are prepared, then no students is left. Find the possible number of students in the class. (a) 60 (b) 72 (c) 80 (d) 96

Options

A. 60
B. 72
C. 80
D. 96

Answer

To find the possible number of students in the class that can be arranged in 6, 8, 12, or 16 rows without any students left over, we need to determine the least common multiple (LCM) of these numbers.

### Step-by-Step Solution:

1. **Identify the numbers**: The numbers we are working with are 6, 8, 12, and 16.

2. **Find the prime factorization of each number**:
- 6 = 2 × 3
- 8 = 2^3
- 12 = 2^2 × 3
- 16 = 2^4

3. **Determine the highest power of each prime factor**:
- For the prime factor 2, the highest power is 2^4 (from 16).
- For the prime factor 3, the highest power is 3^1 (from 6 and 12).

4. **Calculate the LCM**:
- LCM = 2^4 × 3^1 = 16 × 3 = 48

5. **List the multiples of the LCM**:
- The multiples of 48 are: 48, 96, 144, etc.

6. **Check the options given**:
- (a) 60: Not a multiple of 48
- (b) 72: Not a multiple of 48
- (c) 80: Not a multiple of 48
- (d) 96: This is a multiple of 48 (48 × 2)

7. **Conclusion**:
- The only option that is a multiple of 48 is 96. Therefore, the possible number of students in the class is **96**.