If a 9-digit number 32x4115y2 is divisible by 88, then the value of (4x - y) for the smallest possible value of y, is:

Given that,

32x4115y2 is a nine-digit number that is divisible by $$88=11\times 8$$.

The above number will be divisible by 88 if it is individually divisible by 11 and 8.

Divisibility by 8: Any number is always divisible by 8 if the last three-digit number off that particular number is divisible by 8.

so the last three-digit number of the given number is 5y2, so the minimum possible value of y for which it is divisible by 8 =1,5.

Divisibility by 11: Any number is divisible by 11 if the difference between the sum of odd place digit and even place digit is divisible by 11.

Hence, $$2+5+1+x+3-y-1-4-2=4+x-y$$

There is two possible conditions for $$y = 1 and 5$$

i) $$ 4+x-1=3+x $$it will be divisible $$ 11 $$ if $$x=8$$,

ii) $$ 4+x-5=x-1 $$ it will be divisible $$11 $$ if $$x=1$$,

Now,  
If substituting the first values

i) x=8 and y=1 (Here smallest possible value of y)

$$(4x - y) =4\times 8-1=31$$ ( Option A is matching with the answer)

If substituting the second values,

ii) x=1 and y=5 (Largest possible value of Y)

$$(4x-y)=4\times1-5=-1$$ (Not given in the option, so neglecting this.)