If the 8-digit number 179x091y is divisible by 88, then what is the value of (x – y) ?

The correct answer is Option 4 (which is 4).

We are given an 8-digit number $179x091y$ that is divisible by 88. We need to find the value of $(x-y)$.

For a number to be divisible by 88, it must be divisible by both 8 and 11, since 8 and 11 are co-prime factors of 88 (i.e., $88=8\times 11$).

Step 1: Check Divisibility by 8

A number is divisible by 8 if the number formed by its last 3 digits is divisible by 8.
The last 3 digits of $179x091y$ form the number $91y$.

Let's divide 91y by 8:
$91÷8=11$ with a remainder of 3.
So, the last two digits to check with the remainder are $3y$.

For $3y$ to be divisible by 8, the digit $y$ must be 2, because 32 is divisible by 8 ($32=8\times 4$).
Thus, $y=2$.

Step 2: Check Divisibility by 11

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places (starting from the rightmost digit) is either 0 or a multiple of 11.

With $y=2$, the number is $179x0912$.

Sum of digits at odd positions (from right to left):
$2+9+x+7=18+x$

Sum of digits at even positions (from right to left):
$1+0+9+1=11$

Now, calculate the difference:
$\text{Difference}=(18+x)-11=7+x$

For this difference $7+x$ to be a multiple of 11 (and since $x$ is a single digit from 0 to 9):
$7+x=11$
$x=11-7=6$

Step 3: Calculate (x – y)

We found $x=6$ and $y=2$.
$x-y=6-2=4$

Therefore, the value of $(x-y)$ is 4, which corresponds to Option 4.