If x and y are positive integers, is xy even?
(1) x^2 + y^2 - 1 is divisible by 4
(1) x + y is odd
Official Guide question
Answer: D
If x and y are positive integers
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Hi jjjinapinch,
We're told that X and Y are POSITIVE INTEGERS. We're asked if (X)(Y) is EVEN. This is a YES/NO question. This question deals with Number Properties, so you can solve it using those rules and/or by TESTing VALUES.
In this question, it's helpful to consider the possible answers to the given question. The ONLY way for the answer to be "NO" is if BOTH X and Y are ODD. In all other circumstances, the answer to the question will be "YES." You can actually use this knowledge "against" each of the two Facts and eliminate possibilities....
1) X^2 + Y^2 - 1 is divisible by 4
For Fact 1, we can use a Number Property to eliminate a possibility.....
IF.... X and Y are BOTH ODD, then we would have....
(Odd)^2 + (Odd)^2 - 1....
Odd + Odd - Odd = ALWAYS ODD.... but an Odd number CANNOT be divisible by 4, so there is no way that both X and Y can be Odd. This eliminates the possibility of a "No" answer... so the only option that is left is "Yes" (thus, the answer is ALWAYS YES).
Fact 1 is SUFFICIENT
2) X + Y is ODD
For Fact 2, the only way to end up with an ODD sum is if one variable is EVEN and the OTHER is ODD. Since (EVEN)(ODD) = EVEN, the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
We're told that X and Y are POSITIVE INTEGERS. We're asked if (X)(Y) is EVEN. This is a YES/NO question. This question deals with Number Properties, so you can solve it using those rules and/or by TESTing VALUES.
In this question, it's helpful to consider the possible answers to the given question. The ONLY way for the answer to be "NO" is if BOTH X and Y are ODD. In all other circumstances, the answer to the question will be "YES." You can actually use this knowledge "against" each of the two Facts and eliminate possibilities....
1) X^2 + Y^2 - 1 is divisible by 4
For Fact 1, we can use a Number Property to eliminate a possibility.....
IF.... X and Y are BOTH ODD, then we would have....
(Odd)^2 + (Odd)^2 - 1....
Odd + Odd - Odd = ALWAYS ODD.... but an Odd number CANNOT be divisible by 4, so there is no way that both X and Y can be Odd. This eliminates the possibility of a "No" answer... so the only option that is left is "Yes" (thus, the answer is ALWAYS YES).
Fact 1 is SUFFICIENT
2) X + Y is ODD
For Fact 2, the only way to end up with an ODD sum is if one variable is EVEN and the OTHER is ODD. Since (EVEN)(ODD) = EVEN, the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
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Given that x and y are positive integers. We have to determine whether xy is even.jjjinapinch wrote:If x and y are positive integers, is xy even?
(1) x^2 + y^2 - 1 is divisible by 4
(1) x + y is odd
Official Guide question
Answer: D
For xy to be even at least one of x and y must be even. xy is not even if both x and y are odd.
Statement 1: x^2 + y^2 - 1 is divisible by 4.
Since (x^2 + y^2 - 1) is divisible by 4, an even number, (x^2 + y^2 - 1) must be even.
(x^2 + y^2 - 1) --> even
=> x^2 + y^2 --> even + 1 --> even + odd --> odd
If the sum of two positive integers is odd, then each of the positive integers is even.
Thus, x^2 --> even and y^2 ---> even. This follows that x --> even and y --> even.
Thus, xy --> even. Sufficient.
Statement 2: (x + y) is odd.
If the sum of two positive integers is odd, then each of the positive integers is even.
This follows that x --> even and y --> even.
Thus, xy --> even. Sufficient.
The correct answer: D
Hope this helps!
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xy will not be even -- in other words, xy will be ODD -- only if x and y are both odd.jjjinapinch wrote:If x and y are positive integers, is xy even?
(1) x^2 + y^2 - 1 is divisible by 4
(1) x + y is odd
Question stem, rephrased:
Are x and y both odd?
Statement 1: x² + y² - 1 is divisible by 4
x² + y² - 1 = (multiple of 4)
x² + y² = (multiple of 4) + 1
x² + y² = EVEN + ODD
x² + y² = ODD.
The equation in blue implies that x and y CANNOT both be odd, since ODD² + ODD ² = EVEN.
Thus, the answer to the rephrased question stem is NO.
SUFFICIENT.
Statement 2: x + y is odd
The statement above implies that x and y CANNOT both be odd, since ODD + ODD = EVEN.
Thus, the answer to the rephrased question stem is NO.
SUFFICIENT.
The correct answer is D.
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We need to determine whether the product of x and y is even.jjjinapinch wrote:If x and y are positive integers, is xy even?
(1) x^2 + y^2 - 1 is divisible by 4
(1) x + y is odd
Statement One Alone:
x^2 + y^2 − 1 is divisible by 4.
Since 4 is an even number, we need x^2 + y^2 − 1 to be even. In order for x^2 + y^2 − 1 to be even, we need x^2 + y^2 to be odd. If the sum of two squares is odd, one of them must be odd and the other must be even. This means that either x = odd and y = even OR x = even and y = odd. In either case, the product of x and y will be even. Statement one alone is sufficient to answer the question.
Statement Two Alone:
x + y is odd.
Since x + y = odd, either x = odd and y = even OR x = even and y = odd. In either case, the product of x and y will be even. Statement two alone is sufficient to answer the question.
Answer: D
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