Which is an Integer?

IMO D

we can rewrite the expression as

6x + 8y = 33 --> 8y = 33 - 6x ----> 8y = 3(11-2x)

we can see that y = 3 and x = 1.5 are in the line so

PX where P is prime, choosing P = 2 makes PX = 3 which is integer.
QY where Q is even, choosing Q = 2 makes QY = 6 which is an integer.
So II and III could be true.

8y is an even number
so we need 11 - 2x to be an even number so that the product of 3 and 11-2x is an even number.

11-2x has to be even ---> so 2x has to be odd which is impossible unless x is not an integer and therefore x ^ (1/2) cannot be an integer.

Choose D.

You're completely right Bren, I wrongly assumed that Y was an integer complicating the problem.

i can see that a single word "could" has changed the entire question. we can prove all three to be true. what if we restrict the value of x and y to be only integers, does this effect the final answer?