I have a certain number of beads which lie between 600 and 900. If 2 beads are taken away the remainder can be equally divided among 3, 4, 5, 6, 7, or 12 boys. The numbere of beads I have

Options

A. 729
B. 842
C. 576
D. 961

Answer

To solve the problem, we need to find a number of beads (let's call it \( N \)) that lies between 600 and 900. When 2 beads are taken away, the remaining number of beads \( N - 2 \) should be divisible by 3, 4, 5, 6, 7, and 12.

### Step-by-Step Solution:

1. **Identify the Range**:
We know that \( N \) is between 600 and 900. Therefore, we can express this as:
\[ 600 < N < 900 \]

2. **Set Up the Condition**:
When 2 beads are taken away, we have:
\[ N - 2 \]
This number must be divisible by 3, 4, 5, 6, 7, and 12.

3. **Find the LCM**:
To find a number that is divisible by all these numbers, we need to calculate the least common multiple (LCM) of 3, 4, 5, 6, 7, and 12.

- The prime factorization of each number is:
- \( 3 = 3 \)
- \( 4 = 2^2 \)
- \( 5 = 5 \)
- \( 6 = 2 \times 3 \)
- \( 7 = 7 \)
- \( 12 = 2^2 \times 3 \)

- The LCM is found by taking the highest power of each prime:
- \( 2^2 \) from 4 or 12
- \( 3^1 \) from 3, 6, or 12
- \( 5^1 \) from 5
- \( 7^1 \) from 7

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 \]
Calculating this step-by-step:
- \( 4 \times 3 = 12 \)
- \( 12 \times 5 = 60 \)
- \( 60 \times 7 = 420 \)

So, the LCM is \( 420 \).

4. **Determine Possible Values for \( N - 2 \)**:
Since \( N - 2 \) must be a multiple of 420, we can express it as:
\[ N - 2 = 420k \quad \text{for some integer } k \]
Therefore:
\[ N = 420k + 2 \]

5. **Find Suitable \( k \)**:
We need \( N \) to be between 600 and 900:
\[ 600 < 420k + 2 < 900 \]
Subtracting 2 from all parts:
\[ 598 < 420k < 898 \]
Dividing the entire inequality by 420:
\[ \frac{598}{420} < k < \frac{898}{420} \]
Calculating the bounds:
- \( \frac{598}{420} \approx 1.42 \)
- \( \frac{898}{420} \approx 2.14 \)

Thus, \( k \) can be 2 (since \( k \) must be an integer).

6. **Calculate \( N \)**:
Substituting \( k = 2 \):
\[ N = 420 \times 2 + 2 = 840 + 2 = 842 \]

7. **Final Verification**:
We check if \( N - 2 = 840 \) is divisible by 3, 4, 5, 6, 7, and 12:
- \( 840 \div 3 = 280 \)
- \( 840 \div 4 = 210 \)
- \( 840 \div 5 = 168 \)
- \( 840 \div 6 = 140 \)
- \( 840 \div 7 = 120 \)
- \( 840 \div 12 = 70 \)

All results are integers, confirming that \( 840 \) is divisible by all these numbers.

### Conclusion:
The number of beads I have is:
\[ \boxed{842} \]