Are x and y integers?
(1) The product xy is an integer.
(2) x + y is an integer.
OA: E
Tricky question- Are x and y integers?
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Both statements are satisfied by the following cases:Mo2men wrote:Are x and y integers?
(1) The product xy is an integer.
(2) x + y is an integer.
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.
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Dear Mithc,GMATGuruNY wrote:Both statements are satisfied by the following cases:Mo2men wrote:Are x and y integers?
(1) The product xy is an integer.
(2) x + y is an integer.
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.
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?
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In DS, each statement is a PREMISE: a fact that cannot be disputed.Mo2men wrote:√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?
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.
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Thanks Mitch for your response.GMATGuruNY wrote:In DS, each statement is a PREMISE: a fact that cannot be disputed.Mo2men wrote:√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?
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.
1- Does this mean that the DERIVED conclusion equation is wrong?
2- What is your advice when combined statements and choose plug-in values?
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The derived equation is valid in the following sense:Mo2men wrote:1- Does this mean that the DERIVED conclusion equation is wrong?
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.
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.2- What is your advice when combined statements and choose plug-in values?
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).
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