Solving for x - y in a 4-Digit Number Divisible by 11

The problem asks us to find the value of \(x - y\) for a 4-digit number \(1xy7\) that is divisible by 11. To solve this, we will use the divisibility rule for 11.

Understanding the Divisibility Rule for 11

A number is divisible by 11 if the difference between the sum of the digits at the odd places (from the right) and the sum of the digits at the even places (from the right) is either 0 or a multiple of 11. Let's apply this rule to the 4-digit number \(1xy7\).

Applying the Divisibility Rule to 1xy7

In the 4-digit number \(1xy7\):

• The digits at the odd places (1st and 3rd from the right) are 7 and \(x\).
• The digits at the even places (2nd and 4th from the right) are \(y\) and 1.

Now, let's calculate the sum of the digits at the odd places and the sum of the digits at the even places.

• Sum of digits at odd places = \(7 + x\)
• Sum of digits at even places = \(y + 1\)

According to the divisibility rule of 11, the difference between these two sums must be 0 or a multiple of 11.

Difference = (Sum of digits at odd places) - (Sum of digits at even places)

Difference = \((7 + x) - (y + 1)\)

Difference = \(7 + x - y - 1\)

Difference = \(6 + x - y\)

So, for the number \(1xy7\) to be divisible by 11, the value of \(6 + x - y\) must be 0, 11, -11, 22, -22, and so on.

Finding the Possible Values of x - y

\(x\) and \(y\) are digits in a 4-digit number, which means they can take any integer value from 0 to 9.

• The minimum possible value for \(x - y\) is when \(x\) is minimum (0) and \(y\) is maximum (9), which is \(0 - 9 = -9\).
• The maximum possible value for \(x - y\) is when \(x\) is maximum (9) and \(y\) is minimum (0), which is \(9 - 0 = 9\).

Therefore, the possible range for \(x - y\) is from -9 to 9.

Now, let's consider the possible values for \(6 + x - y\) based on the range of \(x - y\):

• Minimum value of \(6 + x - y\) = \(6 + (-9) = -3\)
• Maximum value of \(6 + x - y\) = \(6 + 9 = 15\)

The possible values for \(6 + x - y\) must be multiples of 11 within the range \([-3, 15]\). The only multiple of 11 in this range is 0 or 11.

• Case 1: \(6 + x - y = 0\)
• Case 2: \(6 + x - y = 11\)

Let's solve for \(x - y\) in each case:

• Case 1: \(x - y = 0 - 6 = -6\)
• Case 2: \(x - y = 11 - 6 = 5\)

So, for the number \(1xy7\) to be divisible by 11, the value of \(x - y\) can be either -6 or 5.

Evaluating the Given Options

The given options for the value of \(x - y\) are:

1. -8
2. -2
3. -4
4. -6

Comparing these options with the possible values we found for \(x - y\) (-6 or 5), we see that only -6 is present in the options.

• If \(x - y = -8\), then \(6 + x - y = 6 + (-8) = -2\), which is not a multiple of 11.
• If \(x - y = -2\), then \(6 + x - y = 6 + (-2) = 4\), which is not a multiple of 11.
• If \(x - y = -4\), then \(6 + x - y = 6 + (-4) = 2\), which is not a multiple of 11.
• If \(x - y = -6\), then \(6 + x - y = 6 + (-6) = 0\), which is a multiple of 11.

Thus, the value of \(x - y\) must be -6 for the 4-digit number \(1xy7\) to be divisible by 11.

Therefore, the only value among the given options that makes the 4-digit number \(1xy7\) divisible by 11 is -6.

Revision Table: Divisibility Rules

Understanding divisibility rules is key to solving problems like this. Here's a quick look at some common divisibility rules:

Additional Information: Number Properties

A 4-digit number like \(1xy7\) can be represented using place values. The number can be written as:

\(1 \times 1000 + x \times 100 + y \times 10 + 7 \times 1\)

This representation helps in understanding the contribution of each digit to the total value of the number. When dealing with divisibility rules, especially for numbers like 11, the position of the digits (odd place or even place from the right) is crucial, as seen in the solution above.