If the number x4461 is divisible by 11, what is the face value of x ?

Options

A. 5
B. 2
C. 3
D. 4

Answer

To determine the face value of \( x \) in the number \( x4461 \) such that it is divisible by 11, we will follow these steps:

### Step 1: Identify the positions of the digits
In the number \( x4461 \):
- The digits in odd positions are: \( x \) (1st position), \( 4 \) (3rd position), and \( 1 \) (5th position).
- The digits in even positions are: \( 4 \) (2nd position) and \( 6 \) (4th position).

### Step 2: Calculate the sum of the digits in odd positions
The sum of the digits in odd positions is:
\[ \text{Sum of odd positions} = x + 4 + 1 = x + 5 \]

### Step 3: Calculate the sum of the digits in even positions
The sum of the digits in even positions is:
\[ \text{Sum of even positions} = 4 + 6 = 10 \]

### Step 4: Find the difference between the sums
The difference between the sum of the digits at odd positions and the sum of the digits at even positions is:
\[ \text{Difference} = (x + 5) - 10 = x - 5 \]

### Step 5: Apply the divisibility rule for 11
For a number to be divisible by 11, the difference calculated must be either 0 or a multiple of 11. Thus, we set up the equation:
\[ x - 5 = 0 \quad \text{or} \quad x - 5 = 11 \quad \text{or} \quad x - 5 = -11 \]

### Step 6: Solve for \( x \)
1. From \( x - 5 = 0 \):
\[ x = 5 \]
2. From \( x - 5 = 11 \):
\[ x = 16 \quad \text{(not valid since \( x \) must be a single digit)} \]
3. From \( x - 5 = -11 \):
\[ x = -6 \quad \text{(not valid since \( x \) must be a non-negative digit)} \]

### Conclusion
The only valid solution is \( x = 5 \). Therefore, the face value of \( x \) is:
\[ \boxed{5} \]