It is given that the 10-digit number 55y1725x00 is exactly divisible by 88. What is the value of (y - x) ?

Options

A. 4
B. 3
C. 2
D. 6

Answer

To solve the problem, we need to determine the values of \( y \) and \( x \) such that the 10-digit number \( 55y1725x00 \) is divisible by 88. Since 88 is the product of 8 and 11, we will 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 \( 5x00 \).

To find the value of \( x \), we need to check the divisibility of \( 5x00 \) by 8. We can rewrite \( 5x00 \) as \( 500 + 10x \).

Now, we will check the values of \( x \) from 0 to 9:

- For \( x = 0 \): \( 500 \div 8 = 62.5 \) (not divisible)
- For \( x = 1 \): \( 510 \div 8 = 63.75 \) (not divisible)
- For \( x = 2 \): \( 520 \div 8 = 65 \) (divisible)
- For \( x = 3 \): \( 530 \div 8 = 66.25 \) (not divisible)
- For \( x = 4 \): \( 540 \div 8 = 67.5 \) (not divisible)
- For \( x = 5 \): \( 550 \div 8 = 68.75 \) (not divisible)
- For \( x = 6 \): \( 560 \div 8 = 70 \) (divisible)
- For \( x = 7 \): \( 570 \div 8 = 71.25 \) (not divisible)
- For \( x = 8 \): \( 580 \div 8 = 72.5 \) (not divisible)
- For \( x = 9 \): \( 590 \div 8 = 73.75 \) (not divisible)

Thus, the possible values for \( x \) that make \( 5x00 \) divisible by 8 are \( x = 2 \) and \( x = 6 \).

### Step 2: Check divisibility by 11

Next, we check the divisibility by 11. A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or a multiple of 11.

The digits in odd positions are: \( 5, y, 7, 5, 0 \) (1st, 3rd, 5th, 7th, 9th)
The digits in even positions are: \( 5, 1, 2, x, 0 \) (2nd, 4th, 6th, 8th, 10th)

Calculating the sums:

- Sum of odd positions: \( 5 + y + 7 + 5 + 0 = y + 17 \)
- Sum of even positions: \( 5 + 1 + 2 + x + 0 = x + 8 \)

Now, we find the difference:

\[ (y + 17) - (x + 8) = y - x + 9 \]

For the number to be divisible by 11, \( y - x + 9 \) must be either 0 or a multiple of 11.

### Step 3: Solve for \( y \) and \( x \)

1. If \( x = 2 \):
\[ y - 2 + 9 = y + 7 \]
This must be 0 or a multiple of 11. Therefore, \( y + 7 = 0 \) (not possible since \( y \) is a digit) or \( y + 7 = 11 \) (which gives \( y = 4 \)).

2. If \( x = 6 \):
\[ y - 6 + 9 = y + 3 \]
This must also be 0 or a multiple of 11. Therefore, \( y + 3 = 0 \) (not possible) or \( y + 3 = 11 \) (which gives \( y = 8 \)).

### Final Values

- If \( x = 2 \), then \( y = 4 \).
- If \( x = 6 \), then \( y = 8 \).

### Step 4: Calculate \( y - x \)

1. For \( x = 2 \) and \( y = 4\):
\[ y - x = 4 - 2 = 2 \]

2. For \( x = 6 \) and \( y = 8\):
\[ y - x = 8 - 6 = 2 \]

Thus, in both cases, \( y - x = 2 \).

### Conclusion

The value of \( y - x \) is \( 2 \).