Two numbers 34041 and 32506 when divided by a certain number of three digits, leave the same remainder. The number is :

Options

A. 307
B. 211
C. 247
D. 299

Answer

To solve the problem, we need to find a three-digit number that leaves the same remainder when dividing both 34041 and 32506. The steps to solve this are as follows:

### Step-by-Step Solution:

1. **Identify the Numbers**:
We have two numbers: 34041 and 32506.

2. **Calculate the Difference**:
We find the difference between the two numbers:
\[ 34041 - 32506 = 1535 \]

3. **Find the Factors of the Difference**:
Next, we need to find the factors of 1535. We will do this by checking divisibility with various numbers.

4. **Check for Divisibility**:
- **Divisibility by 5**: Since 1535 ends in 5, it is divisible by 5.
\[ 1535 \div 5 = 307 \]
- **Check if 307 is a three-digit number**: Yes, 307 is a three-digit number.

5. **Verify the Remainder**:
Now we need to check if both 34041 and 32506 leave the same remainder when divided by 307.
- For 34041:
\[ 34041 \div 307 = 110 \quad \text{(quotient)} \]
\[ 34041 - (110 \times 307) = 34041 - 33870 = 171 \quad \text{(remainder)} \]
- For 32506:
\[ 32506 \div 307 = 105 \quad \text{(quotient)} \]
\[ 32506 - (105 \times 307) = 32506 - 32235 = 171 \quad \text{(remainder)} \]

Both numbers leave a remainder of 171 when divided by 307.

6. **Conclusion**:
Therefore, the three-digit number that leaves the same remainder when dividing both 34041 and 32506 is **307**.

### Final Answer:
The number is **307**.

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