Tricky question- Are x and y integers?

[Mo2men]

Are x and y integers?

(1) The product xy is an integer.
(2) x + y is an integer.

OA: E

[GMATGuruNY]

Are x and y integers?

(1) The product xy is an integer.
(2) x + y is an integer.

Both statements are satisfied by the following cases:
Case 1: x=0 and y=0, with the result that xy=0 and x+y=0
In this case, x and y are both integers, so the answer to the question stem is YES.
Case 2: x=√2 and y=-√2, with the result that xy=-2 and x+y=0
In this case, x and y and NOT integers, so the answer to the question stem is NO.
Since the answer is YES in Case 1 but NO in Case 2, the two statements combined are INSUFFICIENT.

The correct answer is E.

[Mo2men]

Are x and y integers?

(1) The product xy is an integer.
(2) x + y is an integer.

Both statements are satisfied by the following cases:
Case 1: x=0 and y=0, with the result that xy=0 and x+y=0
In this case, x and y are both integers, so the answer to the question stem is YES.
Case 2: x=√2 and y=-√2, with the result that xy=-2 and x+y=0
In this case, x and y and NOT integers, so the answer to the question stem is NO.
Since the answer is YES in Case 1 but NO in Case 2, the two statements combined are INSUFFICIENT.

The correct answer is E.

Dear Mithc,
I have tried another approach to prove insufficiency.
Please not that 'i' represents word integer and each color of 'I' means different value.

From Statement 1:
Multiply both sides by...............Nothing will change..The result will be still integer
2xy = Integer = I

From Statement 2:

x +y = Integer = I.....................Square both sides
(x+y)^2 = I^2
x^2 + y^2 + 2xy = I^2
x^2 + y^2 = I^2 - 2xy = Integer = I

From here :
x^2 + y^2 = I
This could be valid if x and y are Integers, for example x =y =2 ...........x^2 + y^2 = 8...........Answer is Yes.
This could be valid if, for example x=y = √2................x^2 + y^2 =4...................................................Answer is No

So Insufficient

Is my approach above is correct? if yes, my example "x=y = √2" works for statement 1 but not 2. However, it works for the DERIVED equation. Can this still be ok, although the example does work for original statement 2?

Thanks

[GMATGuruNY]

√2" works for statement 1 but not 2. However, it works for the DERIVED equation. Can this still be ok, although the example does work for original statement 2?

In DS, each statement is a PREMISE: a fact that cannot be disputed.
The premise in Statement 2 is that x+y = integer.
Your conclusion -- that x²+y² = integer -- is based upon this premise..
√2 violates this premise and thus is not a valid case when the statements are combined.

[Mo2men]

√2" works for statement 1 but not 2. However, it works for the DERIVED equation. Can this still be ok, although the example does work for original statement 2?

In DS, each statement is a PREMISE: a fact that cannot be disputed.
The premise in Statement 2 is that x+y = integer.
Your conclusion -- that x²+y² = integer -- is based upon this premise..
√2 violates this premise and thus is not a valid case when the statements are combined.

Thanks Mitch for your response.

1- Does this mean that the DERIVED conclusion equation is wrong?

2- What is your advice when combined statements and choose plug-in values?

Thanks

[GMATGuruNY]

1- Does this mean that the DERIVED conclusion equation is wrong?

The derived equation is valid in the following sense:
If x and y are values such that their product is an integer and their sum is an integer -- as required by the two statements -- then we can conclude that x²+y² = integer.
x=y=√2 is irrelevant because it violates the condition in red.

The algebra performed in your solution is correct, but it seems time-consuming and not especially helpful.
Even if we correctly deduce that x²+y² = integer, we may still consider only cases that satisfy the two colored conditions above.
It seems more efficient to try to identify cases that satisfy these conditions directly, as in my solution above.

2- What is your advice when combined statements and choose plug-in values?

If your first case yields an answer of YES to the question stem, ask yourself how a second case could yield an answer of NO.
Here, an answer of YES is yielded if x and y are integers.
Subsequent cases should be such that x and y are NOT integers (fractions or roots).