If the seven digit number 5x634y2 is divisible by 88 such that `x!=y`, then ehat is the value of (2y-3x) ?

Options

A. 13
B. 8
C. 15
D. 6

Answer

To solve the problem, we need to determine the values of \( x \) and \( y \) in the seven-digit number \( 5x634y2 \) such that the number is divisible by 88, while also ensuring that \( x \neq y \). The divisibility by 88 requires that the number must be divisible by both 8 and 11.

### Step 1: Check divisibility by 8
To check if a number is divisible by 8, we only need to consider its last three digits. Here, the last three digits are \( 4y2 \).

We will check different values of \( y \) (from 0 to 9) to see which values make \( 4y2 \) divisible by 8.

- If \( y = 0 \): \( 402 \div 8 = 50.25 \) (not divisible)
- If \( y = 1 \): \( 412 \div 8 = 51.5 \) (not divisible)
- If \( y = 2 \): \( 422 \div 8 = 52.75 \) (not divisible)
- If \( y = 3 \): \( 432 \div 8 = 54 \) (divisible)
- If \( y = 4 \): \( 442 \div 8 = 55.25 \) (not divisible)
- If \( y = 5 \): \( 452 \div 8 = 56.5 \) (not divisible)
- If \( y = 6 \): \( 462 \div 8 = 57.75 \) (not divisible)
- If \( y = 7 \): \( 472 \div 8 = 59 \) (divisible)
- If \( y = 8 \): \( 482 \div 8 = 60.25 \) (not divisible)
- If \( y = 9 \): \( 492 \div 8 = 61.5 \) (not divisible)

From this, we find that \( y \) can be either 3 or 7.

### Step 2: Check divisibility by 11
Next, we need to check the divisibility of the number by 11. The rule for divisibility by 11 states that the difference between the sum of the digits in odd positions and the sum of the digits in even positions must be either 0 or divisible by 11.

The digits in odd positions are \( 5, 6, 4, 2 \) and in even positions are \( x, 3, y \).

Calculating the sums:
- Odd positions: \( 5 + 6 + 4 + 2 = 17 \)
- Even positions: \( x + 3 + y \)

The difference is:
\[ |17 - (x + 3 + y)| = |14 - x - y| \]

This difference must be divisible by 11. We will check this for both possible values of \( y \).

#### Case 1: \( y = 3 \)
Substituting \( y = 3 \):
\[ |14 - x - 3| = |11 - x| \]
This can be 0 or 11:
1. \( 11 - x = 0 \) → \( x = 11 \) (not possible)
2. \( 11 - x = 11 \) → \( x = 0 \) (valid)

#### Case 2: \( y = 7 \)
Substituting \( y = 7 \):
\[ |14 - x - 7| = |7 - x| \]
This can be 0 or 11:
1. \( 7 - x = 0 \) → \( x = 7 \) (not valid since \( x \) cannot equal \( y \))
2. \( 7 - x = 11 \) → \( x = -4 \) (not possible)

### Conclusion
The only valid case is:
- \( y = 3 \) and \( x = 0 \)

Now, we need to find the value of \( 2y - 3x \):
\[ 2y - 3x = 2(3) - 3(0) = 6 - 0 = 6 \]

Thus, the final answer is:
\[ \boxed{6} \]