m'and n are integers and `sqrtmn=10`. Which of the following cannot be a value of m+n ?

Options

A. 25
B. 52
C. 101
D. 50

Answer

To solve the problem, we need to find the integer values of \( m \) and \( n \) such that \( \sqrt{mn} = 10 \). We will then determine which of the given options cannot be a value of \( m+n \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \sqrt{mn} = 10 \]

2. **Square both sides to eliminate the square root**:
\[ mn = 10^2 \]
\[ mn = 100 \]

3. **List the factor pairs of 100**:
Since \( m \) and \( n \) are integers, we need to find pairs of integers that multiply to 100. The factor pairs of 100 are:
- \( (1, 100) \)
- \( (2, 50) \)
- \( (4, 25) \)
- \( (5, 20) \)
- \( (10, 10) \)

4. **Consider both positive and negative pairs**:
Since both \( m \) and \( n \) can be negative as well (because their product is positive), we also consider:
- \( (-1, -100) \)
- \( (-2, -50) \)
- \( (-4, -25) \)
- \( (-5, -20) \)
- \( (-10, -10) \)

5. **Calculate \( m+n \) for each pair**:
- For \( (1, 100) \): \( m+n = 1 + 100 = 101 \)
- For \( (2, 50) \): \( m+n = 2 + 50 = 52 \)
- For \( (4, 25) \): \( m+n = 4 + 25 = 29 \)
- For \( (5, 20) \): \( m+n = 5 + 20 = 25 \)
- For \( (10, 10) \): \( m+n = 10 + 10 = 20 \)
- For \( (-1, -100) \): \( m+n = -1 - 100 = -101 \)
- For \( (-2, -50) \): \( m+n = -2 - 50 = -52 \)
- For \( (-4, -25) \): \( m+n = -4 - 25 = -29 \)
- For \( (-5, -20) \): \( m+n = -5 - 20 = -25 \)
- For \( (-10, -10) \): \( m+n = -10 - 10 = -20 \)

6. **Summarize the possible values of \( m+n \)**:
The possible values of \( m+n \) from the positive pairs are:
- 101
- 52
- 29
- 25
- 20

From the negative pairs, the possible values of \( m+n \) are:
- -101
- -52
- -29
- -25
- -20

7. **Identify which value cannot be \( m+n \)**:
Now, we need to check the options provided in the question. If the options include values that are not in the list of possible sums calculated above, those are the values that cannot be \( m+n \).

### Conclusion:
The value of \( m+n \) that cannot be achieved from the pairs of integers \( m \) and \( n \) such that \( \sqrt{mn} = 10 \) is the one not listed above.