Understanding Divisibility by 11

To solve this problem involving the number x4441 and its divisibility by 11, we need to use the divisibility rule for 11. This rule is a handy way to check if a number is divisible by 11 without performing long division.

The rule states that a number is divisible by 11 if the alternating sum of its digits, starting from the rightmost digit, is divisible by 11. This alternating sum can be 0, 11, -11, 22, -22, and so on.

Applying the Divisibility Rule to x4441

The given number is x4441. The digits, starting from the rightmost side, are 1, 4, 4, 4, and x. We need to calculate the alternating sum of these digits:

• Rightmost digit (1st position): +1
• 2nd digit from right (2nd position): -4
• 3rd digit from right (3rd position): +4
• 4th digit from right (4th position): -4
• 5th digit from right (5th position, which is x): +x

The alternating sum is calculated as:

Sum = (Digit at 1st position) - (Digit at 2nd position) + (Digit at 3rd position) - (Digit at 4th position) + (Digit at 5th position)

Substituting the digits from x4441:

Sum = $\(1 - 4 + 4 - 4 + x\)$

Solving for the Face Value of x

Now, let's simplify the expression for the sum:

Sum = $\(1 - 4 + 4 - 4 + x\)$

Sum = $\((1 - 4) + (4 - 4) + x\)$

Sum = $\(-3 + 0 + x\)$

Sum = $\(-3 + x\)$

According to the divisibility rule for 11, this sum ($\(-3 + x\)$) must be divisible by 11. Since x represents a single digit (a face value), it can be any integer from 0 to 9. We need to find a value of x (between 0 and 9) such that $\(-3 + x\)$ is a multiple of 11.

Possible multiples of 11 are 0, $\(\pm 11\)$, $\(\pm 22\)$, etc.

• If $\(-3 + x = 0\)$, then $\(x = 3\)$. This is a single digit (0-9).
• If $\(-3 + x = 11\)$, then $\(x = 14\)$. This is not a single digit.
• If $\(-3 + x = -11\)$, then $\(x = -8\)$. This is not a single digit.

The only valid single-digit value for x that makes $\(-3 + x\)$ divisible by 11 is when the sum is 0, which gives $\(x = 3\)$.

Therefore, the face value of x is 3.

Verification

If x = 3, the number is 34441. Let's check the alternating sum of its digits:

$\(1 - 4 + 4 - 4 + 3 = -3 + 4 - 4 + 3 = 1 - 4 + 3 = -3 + 3 = 0\)$

Since the alternating sum is 0, which is divisible by 11, the number 34441 is indeed divisible by 11. This confirms that the face value of x is 3.

Additional Information: Face Value and Place Value

In a number, each digit has a face value and a place value.

• Face Value: The face value of a digit is simply the digit itself, regardless of its position in the number. For example, in the number 34441, the face value of the digit '3' is 3, and the face value of the digit '4' is 4, and the face value of the digit '1' is 1.
• Place Value: The place value of a digit depends on its position in the number. It is the face value multiplied by the value of the position (ones, tens, hundreds, thousands, etc.). In the number 34441:
  • The place value of 1 is $\(1 \times 1 = 1\)$ (ones place).
  • The place value of the first 4 (from the right) is $\(4 \times 10 = 40\)$ (tens place).
  • The place value of the second 4 is $\(4 \times 100 = 400\)$ (hundreds place).
  • The place value of the third 4 is $\(4 \times 1000 = 4000\)$ (thousands place).
  • The place value of 3 is $\(3 \times 10000 = 30000\)$ (ten thousands place).

The question asked for the face value of x, which is simply the digit itself.