For what value of `**` the number `56^(**) 891` will be perfectly divisible by 11 ?

Options

A. 1
B. 3
C. 5
D. 7

Answer

To determine the value of `**` (let's denote it as `x`) such that the number `56x891` is perfectly divisible by 11, we can use the divisibility rule for 11. According to this rule, the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions must be a multiple of 11.

### Step-by-Step Solution:

1. **Identify the positions of the digits**:
- The number is `56x891`.
- The positions of the digits are:
- 1st position (odd): 5
- 2nd position (even): 6
- 3rd position (odd): x
- 4th position (even): 8
- 5th position (odd): 9
- 6th position (even): 1

2. **Sum the digits in odd positions**:
- Odd positions: 5 (1st) + x (3rd) + 9 (5th) = 5 + x + 9 = x + 14

3. **Sum the digits in even positions**:
- Even positions: 6 (2nd) + 8 (4th) + 1 (6th) = 6 + 8 + 1 = 15

4. **Set up the equation for divisibility by 11**:
- According to the rule, we need:
\[ |(x + 14) - 15| \text{ must be a multiple of } 11 \]
- Simplifying this gives:
\[ |x - 1| \text{ must be a multiple of } 11 \]

5. **Consider possible values for x**:
- Since `x` is a single digit (0 to 9), we can evaluate:
- If \( x - 1 = 0 \) → \( x = 1 \)
- If \( x - 1 = 11 \) → \( x = 12 \) (not valid since x must be a single digit)
- If \( x - 1 = -11 \) → \( x = -10 \) (not valid since x must be a single digit)

6. **Conclusion**:
- The only valid solution is \( x = 1 \).
- Therefore, the value of `**` that makes `56**891` divisible by 11 is **1**.