I'm happy to help with this.

First of all, this is a very challenging GMAT problem --- it really demands a great deal of number sense to answer this efficiently.

The prompt tells us x & y are positive integer (mercifully!) and x*y is divisible by 4. That could happen if either one of them is a multiple of four, or if both of them are even.

(A) Counter-example: If x = 2, then y could not be odd, because 2*(odd) will never be divisible by four. NOT ALWAYS TRUE.
(B) If x is odd, then there are no factors of 2 in x. There are two factors of 2 in x*y, since x*y is be divisible by 4. If those two factors of 2 don't come from x, they must come from y. Therefore y MUST BE divisible by four.
(C) Counter-example: x = 3 and y = 12 --- x + 12 = 15 is odd, but y/x = 12/3 = 4 is an integer. NOT ALWAYS TRUE.
(D) Counter-example: x = 12 and y = 8 ---- x + y = 20 is even, but x/y = 12/8 = 3/2 is not an integer. NOT ALWAYS TRUE.
(E) Counter-example: x = 3 and y = 4 ---- x^y = 3^4 = 81 is not even. NOT ALWAYS TRUE.

In every pair in the counter-examples, the numbers were chosen so that x*y is divisible by 4, and any additional condition in that particular choice is met.

This blog is tangentially related -- it may help a bit with number sense:
http://magoosh.com/gmat/2012/gmat-math- ... variables/

Does all this make sense?

Mike
_________________

Mike McGarry
Magoosh Test Prep

If xy meant to be two-digit integer xy, then it would be mentioned something like "x and y are digits of two-digit integer xy".

If x and y are positive integer and xy is divisible by 4, which of the following must be true?

A. If x is even then y is odd.
B. If x is odd then y is a multiple of 4.
C. If x+y is odd then \frac{y}{x} is not an integer.
D. If x+y is even then \frac{x}{y} is an integer.
E. x^y is even.

Notice that the question asks which of the following MUST be true not COULD be true.

A. If x is even then y is odd --> not necessarily true, consider: x=y=2=even;

B. If x is odd then y is a multiple of 4 --> always true: if x=odd then in order xy to be a multiple of 4 y mst be a multiple of 4;

C. If x+y is odd then \frac{y}{x} is not an integer --> not necessarily true, consider: x=1 and y=4;

D. If x+y is even then \frac{x}{y} is an integer --> not necessarily true, consider: x=2 and y=4;

E. x^y is even --> not necessarily true, consider: x=1 and y=4;

Answer: B.
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Dear gmatchase

First of all, I'd like to suggest --- don't over-do the "math" notation. That's useful for fractions & exponents, but for ordinary x+y, we don't need the special notation.

The blanket assumption throughout all of algebra is that writing two variables next to each other (mx) or writing a number directly next to a variable (5y) always means multiplication. That's the meaning 100% of the time, unless something different is very explicitly specified. If the problem wanted the variables to represent digits, first of all, it probably would have used letters from the beginning of the alphabet, certainly not x & y (the standard algebraic variables), and it would have to say, quite explicitly, something like "Let a and b be digits in a the two-digit number ab." In other words, because the blank assumption is --- two variables next to each other always means multiply --- when a problem wants to indicate digits or anything else, it practically has to set up traffic cones and flashing signs to indicate the change. For a digits problem, it would have to explain, explicitly, what each variable represented. Any problem that just tosses an xy at you with no verbiage will never be using those variable as digits --- that xy will be multiplication 100% of the time.

Here's an example of a rare digit problem.
http://gmat.magoosh.com/questions/54
When you submit you answer, the following page will have a full video explanation.

Does all this make sense?

Mike
_________________

Mike McGarry
Magoosh Test Prep