Theory

➡ GCD of two numbers is always smaller than or equal to the smaller of those two numbers
➡ GCD (a,b) <= Smaller (a,b)

Which of the following CANNOT be the greatest common divisor of two positive integers x and y

Let's take each option choice and evaluate

(A) 1
Now, we can take co-prime values of x and y (co-primes are numbers which have only 1 as as the common factor) and this will be true.
Ex, x=2 and y=3 => GCD = 1
=> TRUE

(B) x
This can be true when one number is x and other number is a multiple of x
Example: One number is 2 (which is x) and other number is 2*2 = 4 (which is a multiple of x). Making the GCD = 2 (which is x)
=> TRUE

(C) y
This can be true when one number is y and other number is a multiple of y
Example: One number is 2 (which is y) and other number is 2*2 = 4 (which is a multiple of y). Making the GCD = 2 (which is y)
=> TRUE

(D) x - y
Take x = 4, y = 2.
x - y = 4-2 = 2
GCD (4,2) = 2
=> TRUE

(E) x + y
Now, GCD of two numbers is always smaller than or equal to the smaller of those two numbers
=> GCD(x,y) <= Smaller of x and y
So, GCD can never be x+y as x+y will be greater than both x and y.
=> FALSE

So, Answer will be E
Hope it helps!

To learn more about LCM and GCD watch the following videos

x+y is greater than both x and y so it can't be a divisor of either of them: