Without much thought and hence I may be wrong . I think the information provided by the statements is insufficient because it is stated "non-zero integers" and there will always be a combination of positive X & Y and a combination of negative X & Y and hence the statements are insufficient

I'll give it a try:

1. When x is divided by 2y, the remainder is 4.
=> X = (Multiple of 2y) + 4
=> X = (Multiple of y) + 4
=> When x is divided by y, the remainder is 4.

2. When x +y is divided by y, the remainder is 4.
(x+y)/y = x/y+y/y
y/y will have remainder of 0, so the remainder 4 is from x/y

I feel each statement is sufficient on its own.

from 1 ,
x=2ny+4 ..n is any integer
when x is divided by y reminder is 4/y..now if 4 is divisible by y reminder will be 0
if 4 is divisible by y values of y can be 1,2,4

among these only for y=4 4 can be reminder when x is divided with 2y but reminder is 0 when x is divided with y...not sufficient

From 2
When x +y is divided by y, the remainder is 4
since y is completely divisible by y itself the reminder comes from x i.e when x is divided by y the reminder is 4..sufficient

Ans option B
what is OA?