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Location: Canada
If x² + y³ + 3x 2y + xy + 1 is odd, is x + y odd?
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24 Aug 2022, 07:53
24 Aug 2022, 07:53
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If 2x² + y³ + 3x – 2y + xy + 1 is even, is x + y odd?
(1) xy is even
(2) y is even
(1) xy is even
(2) y is even
Re: If x² + y³ + 3x 2y + xy + 1 is odd, is x + y odd?
[#permalink]
24 Aug 2022, 20:42
24 Aug 2022, 20:42
2x² + y³ + 3x – 2y + xy + 1 = Even
2x² and 2y will be even
Therefore, E + y³ + 3x - E + xy + 1 = Even
E + y³ + 3x + xy + 1 = Even (E - E = E)
Odd + y³ + 3x + xy = Even (E + 1 = Odd)
That means, y³ + 3x + xy has to be Odd. With this information, lets look at Statement 1.
Statement 1 - xy is even
that means either x or Y is even or both can be even
We know that y³ + 3x + xy has to be Odd therfore both can't be even
Therefore, either x or y has to be even and the other on has to be odd
x + y = E + O = O
x + Y is not Even and statement 1 is sufficent to answer the question
Statement 2 - y is even
We know that y³ + 3x + xy = Odd
If y is even then y³ and xy will be even that is E + 3x + E = Odd`
E + 3x = Odd. So, 3x has to be odd, which means x has to be odd
and form statement 2 we know y is even
x + y = O + E = O
x + Y is not Even and statement 2 is sufficent to answer the question
Hence. answer is (D)
2x² and 2y will be even
Therefore, E + y³ + 3x - E + xy + 1 = Even
E + y³ + 3x + xy + 1 = Even (E - E = E)
Odd + y³ + 3x + xy = Even (E + 1 = Odd)
That means, y³ + 3x + xy has to be Odd. With this information, lets look at Statement 1.
Statement 1 - xy is even
that means either x or Y is even or both can be even
We know that y³ + 3x + xy has to be Odd therfore both can't be even
Therefore, either x or y has to be even and the other on has to be odd
x + y = E + O = O
x + Y is not Even and statement 1 is sufficent to answer the question
Statement 2 - y is even
We know that y³ + 3x + xy = Odd
If y is even then y³ and xy will be even that is E + 3x + E = Odd`
E + 3x = Odd. So, 3x has to be odd, which means x has to be odd
and form statement 2 we know y is even
x + y = O + E = O
x + Y is not Even and statement 2 is sufficent to answer the question
Hence. answer is (D)
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Re: If x² + y³ + 3x 2y + xy + 1 is odd, is x + y odd?
[#permalink]
25 Aug 2022, 07:58
25 Aug 2022, 07:58
Expert Reply
Top Contributor
BrentGMATPrepNow wrote:
If 2x² + y³ + 3x – 2y + xy + 1 is even, is x + y odd?
(1) xy is even
(2) y is even
(1) xy is even
(2) y is even
Answer:
Target question: Is x + y odd?
Given: 2x² + y³ + 3x – 2y + xy + 1 is even
STRATEGY: When the given information in a data sufficiency question yields a limited number of possible cases (5 or fewer), you can save make things easier by listing those possible cases before analyzing each statement.
For the given information, we need only test 4 possible cases (i.e., x and y are both even, x is even and y is odd, etc.)
To test each case, I like to plug in 0 for even numbers and 1 for odd numbers.
When I do this, I get:
So, there are exactly 3 cases in which 2x² + y³ + 3x – 2y + xy + 1 is even.
I'm now ready for the statements!!!
Statement 1: xy is even
xy is even in cases i and ii
Since the answer to the target question (Is x + y odd?) is YES for cases i and ii, it MUST be the case that x + y is odd
So, statement 1 is SUFFICIENT
Statement 2: y is even
y is even in only one of the three possible cases (case ii), which means it MUST be the case that x + y is odd
So, statement 2 is SUFFICIENT
Answer: D
