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 can follow these steps:

### Step 1: Understand the divisibility rule for 11
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 a multiple of 11.

### Step 2: Identify the positions of the digits
In the number \( x4461 \):
- Odd position digits: \( x \) (1st), 4 (3rd), 1 (5th)
- Even position digits: 4 (2nd), 6 (4th)

### Step 3: Calculate the sums
- Sum of the digits in odd positions:
\[ S_{\text{odd}} = x + 4 + 1 = x + 5 \]
- Sum of the digits in even positions:
\[ S_{\text{even}} = 4 + 6 = 10 \]

### Step 4: Set up the equation for divisibility
According to the divisibility rule:
\[ |S_{\text{odd}} - S_{\text{even}}| = |(x + 5) - 10| = |x - 5| \]
This must be either 0 or a multiple of 11.

### Step 5: Solve for \( x \)
1. **Case 1:** \( |x - 5| = 0 \)
- This gives us \( x - 5 = 0 \) or \( x = 5 \).

2. **Case 2:** \( |x - 5| = 11 \)
- This gives us two equations:
- \( x - 5 = 11 \) → \( x = 16 \) (not valid since \( x \) must be a single digit)
- \( x - 5 = -11 \) → \( x = -6 \) (not valid since \( x \) must be a non-negative digit)

### Conclusion
The only valid solution is \( x = 5 \).

### Final Answer
The face value of \( x \) is \( 5 \).

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