If m and n are integers then is n completely divisible by 10?
A. The value of `"("(m)/(10)+(n)/(10)")"` is a integer value.
B. The value of `"("(m)/(7)+(n)/(10)")"` is an integer value.
C. value of n is greater than m.

Options

A. Any two of them
B. A and B together
C. Any of them
D. All statements are required

Explanation

To determine if \( n \) is completely divisible by 10, we need to analyze the statements provided:

### Step-by-Step Solution:

1. **Understanding the Statements**:
- **Statement A**: The value of \( \left( \frac{m}{10} + \frac{n}{10} \right) \) is an integer.
- **Statement B**: The value of \( \left( \frac{m}{7} + \frac{n}{10} \right) \) is an integer.
- **Statement C**: The value of \( n \) is greater than \( m \).

2. **Analyzing Statement A**:
- If \( \left( \frac{m}{10} + \frac{n}{10} \right) \) is an integer, then \( \frac{m+n}{10} \) must also be an integer.
- This implies that \( m+n \) is divisible by 10, or \( m+n = 10k \) for some integer \( k \).

3. **Analyzing Statement B**:
- If \( \left( \frac{m}{7} + \frac{n}{10} \right) \) is an integer, then \( \frac{10m + 7n}{70} \) must also be an integer.
- This implies that \( 10m + 7n \) is divisible by 70, or \( 10m + 7n = 70i \) for some integer \( i \).

4. **Combining Statements A and B**:
- From Statement A, we have \( m+n = 10k \).
- From Statement B, we can rearrange \( 10m + 7n = 70i \) to express \( n \) in terms of \( m \):
\[ 7n = 70i - 10m \implies n = 10i - \frac{10m}{7} \]
- Since \( n \) must be an integer, \( \frac{10m}{7} \) must also be an integer, which implies that \( m \) must be a multiple of 7.

5. **Finding \( n \) in Terms of \( k \)**:
- From \( m+n = 10k \), we can express \( n \) as:
\[ n = 10k - m \]
- Substituting \( m = 7j \) (where \( j \) is an integer) into the equation gives:
\[ n = 10k - 7j \]
- This means \( n \) can be expressed as a linear combination of integers, and we need to check if \( n \) is divisible by 10.

6. **Conclusion**:
- For \( n \) to be completely divisible by 10, we need both statements A and B to hold true.
- Statement C does not provide any information about the divisibility of \( n \) by 10.

### Final Answer:
- Therefore, the answer is that both **Statements A and B** are necessary to conclude that \( n \) is completely divisible by 10.