Concept: The questions tests on basics of Number Properties
We need to remember that, in such questions the GMAT expects you to use a mathematical operation instead of just plugging values. It is also good to remember that in questions on Odds & Evens, GMAT tests whether you can convert addition/subtraction of terms to multiplication of terms, because multiplication of Odds & Evens is easier to work with.
For example, if two numbers are added and you get an odd number, you will have to deal with two cases; on the other hand, when you multiply two numbers and get an odd number, there can only be one conclusion – that both are odd.
Therefore, when you see a question on odds and evens, try to convert the expression to a multiplication of terms by factoring out common terms.
Question data tells us that x, y and z are integers and xy + z = odd.
From statement I alone, xy + xz = even. Note how there’s an xy in both equations. We can use Algebra to deal with these two equations:
xy + z = odd
xy + xz = even
Subtracting the first equation from the second, xy gets cancelled leaving us with xz – z = even – odd.
Factoring out z, we have z(x-1) = odd. This means both z and (x-1) are odd numbers. (x-1) can be an odd number only when x is an even number.
Is x even? It certainly is. Statement I alone is sufficient to give a definite YES.
Answer options B, C and E can be eliminated. Possible answer options are A or D.
From statement II alone, y + xz = odd. The two equations we now have are:
y + xz = odd
xy + z = odd.
Adding these two equations, we have,
xy + y + z + xz = odd + odd
Factoring out the common terms and simplifying, we have,
y(x+1) + z(x+1) = even.
Since (x+1) is common, we can factor it out and rewrite the equation as
(x+1) (y+z) = even.
Now, when the product of two numbers is even, either of them could be even or both could be even. Therefore, the above equation is not sufficient to conclude if (x+1) is odd or even.
If (x+1) = odd, x = even; but, if (x+1) = even, x = odd.
Statement II alone is insufficient to give a definite YES or NO. Answer option D can be eliminated.
The correct answer option is A.