If a number 657423547 X 46 is divisible by 11, then find the value of X.

Options

A. 7
B. 9
C. 8
D. 6

Answer

To determine the value of \( X \) in the number \( 657423547X46 \) such that it is divisible by 11, we can 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 at odd positions and the sum of the digits at even positions is either 0 or a multiple of 11.

### Step-by-Step Solution:

1. **Identify the positions of the digits**:
The number is \( 6, 5, 7, 4, 2, 3, 5, 4, 7, X, 4, 6 \). We will label the positions from left to right:
- Odd positions: 1st, 3rd, 5th, 7th, 9th, 11th (6, 7, 2, 5, 7, 4)
- Even positions: 2nd, 4th, 6th, 8th, 10th, 12th (5, 4, 3, 4, X, 6)

2. **Calculate the sum of the digits at odd positions**:
\[ \text{Sum of odd positions} = 6 + 7 + 2 + 5 + 7 + 4 = 31 \]

3. **Calculate the sum of the digits at even positions**:
\[ \text{Sum of even positions} = 5 + 4 + 3 + 4 + X + 6 = 22 + X \]

4. **Find the difference between the sums**:
\[ \text{Difference} = \text{Sum of odd positions} - \text{Sum of even positions} = 31 - (22 + X) = 31 - 22 - X = 9 - X \]

5. **Set the difference to be a multiple of 11**:
For the number to be divisible by 11, the difference \( 9 - X \) must be 0 or a multiple of 11. The possible values for \( 9 - X \) can be:
- \( 9 - X = 0 \) which gives \( X = 9 \)
- \( 9 - X = 11 \) which gives \( X = -2 \) (not a valid digit)
- \( 9 - X = -11 \) which gives \( X = 20 \) (not a valid digit)

The only valid solution is \( X = 9 \).

6. **Conclusion**:
Thus, the value of \( X \) that makes the number \( 657423547X46 \) divisible by 11 is:
\[ \boxed{9} \]