If a number 6784329x145 is divisible by 11, then find the value of x.

Options

A. 3
B. 4
C. 5
D. 7

Answer

To determine the value of \( x \) such that the number \( 6784329x145 \) is divisible by 11, we will use the divisibility rule for 11. According to this rule, a number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or divisible by 11.

### Step-by-Step Solution:

1. **Identify the positions of the digits**:
- The digits of the number \( 6784329x145 \) can be indexed as follows:
- 1st (odd): 6
- 2nd (even): 7
- 3rd (odd): 8
- 4th (even): 4
- 5th (odd): 3
- 6th (even): 2
- 7th (odd): 9
- 8th (even): \( x \)
- 9th (odd): 1
- 10th (even): 4
- 11th (odd): 5

2. **Calculate the sum of the digits in odd positions**:
- Odd positions: \( 6, 8, 3, 9, 1, 5 \)
- Sum = \( 6 + 8 + 3 + 9 + 1 + 5 = 32 \)

3. **Calculate the sum of the digits in even positions**:
- Even positions: \( 7, 4, 2, x, 4 \)
- Sum = \( 7 + 4 + 2 + x + 4 = 17 + x \)

4. **Find the difference between the two sums**:
- Difference = \( |32 - (17 + x)| = |32 - 17 - x| = |15 - x| \)

5. **Set up the condition for divisibility by 11**:
- For the number to be divisible by 11, the difference \( |15 - x| \) must be either 0 or a multiple of 11.
- This gives us two cases to consider:
- Case 1: \( 15 - x = 0 \)
- Case 2: \( 15 - x = 11 \)

6. **Solve for \( x \) in both cases**:
- **Case 1**:
- \( 15 - x = 0 \)
- \( x = 15 \) (not valid since \( x \) must be a single digit)

- **Case 2**:
- \( 15 - x = 11 \)
- \( 15 - 11 = x \)
- \( x = 4 \)

7. **Conclusion**:
- The value of \( x \) that makes the number \( 6784329x145 \) divisible by 11 is \( x = 4 \).