If a nine-digit number 43x1145y2 is divisible by 88, then the value of (2x-y), for the largest value of y, is:

Options

A. 1
B. 5
C. 0
D. `- 1`

Answer

To determine the value of \(2x - y\) for the largest value of \(y\) in the nine-digit number \(43x1145y2\) that is divisible by \(88\), we need to check the divisibility by both \(8\) and \(11\).

### Step 1: Check divisibility by 8
A number is divisible by \(8\) if its last three digits form a number that is divisible by \(8\). The last three digits of our number are \(5y2\).

To find the values of \(y\) that make \(5y2\) divisible by \(8\), we will evaluate \(5y2\) for \(y = 0, 1, 2, \ldots, 9\):

- \(y = 0\): \(502 \div 8 = 62.75\) (not divisible)
- \(y = 1\): \(512 \div 8 = 64\) (divisible)
- \(y = 2\): \(522 \div 8 = 65.25\) (not divisible)
- \(y = 3\): \(532 \div 8 = 66.5\) (not divisible)
- \(y = 4\): \(542 \div 8 = 67.75\) (not divisible)
- \(y = 5\): \(552 \div 8 = 69\) (divisible)
- \(y = 6\): \(562 \div 8 = 70.25\) (not divisible)
- \(y = 7\): \(572 \div 8 = 71.5\) (not divisible)
- \(y = 8\): \(582 \div 8 = 72.75\) (not divisible)
- \(y = 9\): \(592 \div 8 = 74\) (divisible)

From this, the possible values of \(y\) that make \(5y2\) divisible by \(8\) are \(1, 5, 9\). The largest value is \(y = 9\).

### Step 2: Check divisibility by 11
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 divisible by \(11\).

The digits of the number \(43x1145y2\) are positioned as follows:
- Odd positions: \(4, x, 1, 5, 2\) (sum = \(4 + x + 1 + 5 + 2 = x + 12\))
- Even positions: \(3, 1, 4, y\) (sum = \(3 + 1 + 4 + y = 8 + y\))

Now, we calculate the difference:
\[ (x + 12) - (8 + y) = x + 4 - y \]
We need \(x + 4 - y\) to be divisible by \(11\).

### Step 3: Substitute \(y = 9\)
Substituting \(y = 9\):
\[ x + 4 - 9 = x - 5 \]
We need \(x - 5\) to be divisible by \(11\). The possible values for \(x\) can be \(5\) (as \(x\) must be a single digit).

### Step 4: Calculate \(2x - y\)
Now substituting \(x = 5\) and \(y = 9\):
\[ 2x - y = 2(5) - 9 = 10 - 9 = 1 \]

### Final Answer
Thus, the value of \(2x - y\) for the largest value of \(y\) is:
\[ \boxed{1} \]