Consider the following statements :
A number `a_(1)a_(2)a_(3)a_(4)a_(5)a_(6)` is divisible by 11 if
1. `(a_(1)+a_(3)+a_(5))-(a_(2)+a_(4)+a_(6))=0`
2. `(a_(1)+a_(3)+a_(5))-(a_(2)+a_(4)+a_(6))` is divisible by 11
Which of these statements is/are correct?

Options

A. 1 alone
B. 2 alone
C. Both 1 and 2
D. Neither 1 nor 2

Answer

To determine which of the statements regarding divisibility by 11 are correct, we will analyze both statements step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to check the validity of two statements about a number represented as `a1a2a3a4a5a6` (where `a1`, `a2`, `a3`, `a4`, `a5`, and `a6` are its digits) being divisible by 11.

2. **Statement 1**:
The first statement claims that a number is divisible by 11 if:
\[ (a1 + a3 + a5) - (a2 + a4 + a6) = 0 \]
This means that the sum of the digits at odd positions (1st, 3rd, 5th) should equal the sum of the digits at even positions (2nd, 4th, 6th).

3. **Statement 2**:
The second statement claims that a number is divisible by 11 if:
\[ (a1 + a3 + a5) - (a2 + a4 + a6) \text{ is divisible by } 11 \]
This means that the difference between the sum of the digits at odd positions and the sum of the digits at even positions should be a multiple of 11.

4. **Analyzing Statement 1**:
- If the difference is 0, it is indeed a multiple of 11 (since 0 is divisible by every integer).
- Therefore, if Statement 1 is true, Statement 2 must also be true because 0 is divisible by 11.

5. **Analyzing Statement 2**:
- If the difference is divisible by 11, it covers cases where the difference could be 0 or any other multiple of 11.
- Thus, Statement 2 is also correct.

6. **Conclusion**:
Both statements are correct. A number is divisible by 11 if either the difference of the sums of its odd and even positioned digits is zero or if that difference is divisible by 11.

### Final Answer:
Both statements are correct.