67x19 is a multiple of 11. Find all possible values of digit x.

Answer

To solve the problem of finding the possible values of the digit \( x \) in the number \( 67x19 \) such that it is a multiple of 11, we will use the divisibility rule for 11.

### Step-by-step Solution:

1. **Understanding the divisibility rule for 11**:
A number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either 0 or a multiple of 11.

2. **Identifying the positions of the digits**:
For the number \( 67x19 \):
- Odd positions: \( 6 \) (1st), \( x \) (3rd), \( 9 \) (5th)
- Even positions: \( 7 \) (2nd), \( 1 \) (4th)

3. **Calculating the sums**:
- Sum of digits in odd positions: \( 6 + x + 9 = x + 15 \)
- Sum of digits in even positions: \( 7 + 1 = 8 \)

4. **Setting up the equation**:
According to the divisibility rule:
\[ |(x + 15) - 8| = |x + 7| \]
This difference must be a multiple of 11.

5. **Finding possible values of \( x \)**:
We need to check for values of \( x \) from 0 to 9:
- \( |x + 7| = 0 \) or \( |x + 7| = 11 \)

**Case 1**: \( x + 7 = 0 \)
- \( x = -7 \) (not valid since \( x \) must be a digit)

**Case 2**: \( x + 7 = 11 \)
- \( x = 4 \) (valid digit)

**Case 3**: \( x + 7 = -11 \)
- \( x = -18 \) (not valid)

**Case 4**: \( x + 7 = 11 \) (considering negative)
- \( x = 4 \) (already found)

**Case 5**: \( x + 7 = 0 \) (considering negative)
- \( x = -7 \) (not valid)

6. **Verifying the found value**:
We only found \( x = 4 \). Now we need to check if \( 67x19 \) is divisible by 11 when \( x = 4 \):
- The number becomes \( 67419 \).
- Odd positions: \( 6 + 4 + 9 = 19 \)
- Even positions: \( 7 + 1 = 8 \)
- Difference: \( 19 - 8 = 11 \) (which is divisible by 11)

### Conclusion:
The only possible value of the digit \( x \) such that \( 67x19 \) is a multiple of 11 is \( x = 4 \).