What is the greatest common factor of x and y ?
(1) x and y are both divisible by 4 --> clearly insufficient: of course 4 itself could be the GCD but it's also possible that x and y share some other common factor more than 4, for example 5 (in this case both will be divisible by 20), or 8, or 4,000,000 ... so GCD could be more than 4 as well. Though from this statement we know that GCD cannot possibly be less than 4.
(2) x - y = 4 --> according to the property #2: GCD(x,y)≤4
GCD(x,y)≤4, but still insufficient.
(1)+(2) From (1) GCD≥4
GCD≥4 and from (2) GCD≤4
GCD≤4 --> GCD=4
GCD=4 Sufficient.
Answer: C.