If a number 5x423y is completely divisible by 88 then find the value of 5x - 8y.

Options

A. 16
B. 24
C. 32
D. 40

Answer

To solve the problem, we need to determine the values of \( x \) and \( y \) in the number \( 5x423y \) such that it is divisible by 88. Since \( 88 = 11 \times 8 \), we will check the divisibility by both 8 and 11.

### Step-by-step Solution:

1. **Check divisibility by 8**:
- For a number to be divisible by 8, the last three digits must be divisible by 8. Here, the last three digits are \( 423y \).
- We need to check \( 23y \) (the last three digits) for divisibility by 8.
- We can test values of \( y \) from 0 to 9:
- \( y = 0 \): \( 230 \div 8 = 28.75 \) (not divisible)
- \( y = 1 \): \( 231 \div 8 = 28.875 \) (not divisible)
- \( y = 2 \): \( 232 \div 8 = 29 \) (divisible)
- \( y = 3 \): \( 233 \div 8 = 29.125 \) (not divisible)
- \( y = 4 \): \( 234 \div 8 = 29.25 \) (not divisible)
- \( y = 5 \): \( 235 \div 8 = 29.375 \) (not divisible)
- \( y = 6 \): \( 236 \div 8 = 29.5 \) (not divisible)
- \( y = 7 \): \( 237 \div 8 = 29.625 \) (not divisible)
- \( y = 8 \): \( 238 \div 8 = 29.75 \) (not divisible)
- \( y = 9 \): \( 239 \div 8 = 29.875 \) (not divisible)
- The only value of \( y \) that makes \( 23y \) divisible by 8 is \( y = 2 \).

2. **Check divisibility by 11**:
- For a number to be divisible by 11, the difference between the sum of the digits at odd positions and the sum of the digits at even positions must be 0 or a multiple of 11.
- The digits in \( 5x4232 \) are:
- Odd positions: \( 5, 4, 3 \) → Sum = \( 5 + 4 + 3 = 12 \)
- Even positions: \( x, 2, 2 \) → Sum = \( x + 2 + 2 = x + 4 \)
- The difference is \( 12 - (x + 4) = 8 - x \).
- For divisibility by 11, \( 8 - x \) should be 0 or a multiple of 11. The only feasible solution is:
- \( 8 - x = 0 \) → \( x = 8 \).

3. **Calculate \( 5x - 8y \)**:
- Now we have \( x = 8 \) and \( y = 2 \).
- Substitute these values into \( 5x - 8y \):
\[ 5x - 8y = 5(8) - 8(2) = 40 - 16 = 24. \]

### Final Answer:
The value of \( 5x - 8y \) is \( 24 \).