If x and y are integers, is x + y even?

Question

(1) x + 2y is odd.
(2) xy is odd.

emanresu1 · Answer

This one (from GMATPrep Exam 5) has me pulling my hair out. According to the OA, only (2) is sufficient. I think (1) is sufficient alone too, however.

My logic for (1):
- x + 2y -> O
- 2y must be even (any integer times 2 is even)
-  thus x must be odd 
- "x + y" is only odd if either x or y is odd
-  thus (1) indicates that "x + y" is odd

Nikhil · Answer

To determine whether x + y is even or odd

Statement 1

x + 2y is odd

=> 2y is always even irrespective of even/odd nature of y (any integer multiplied by even results in an even integer)

=> x + 2y = odd

=> x = odd - 2y = odd - even = odd

So, x is odd and y can either be odd or even

if x is odd and y is odd => x + y = odd + odd = even

if x is odd and y is even => x + y = odd + even = odd

As we have two possible cases, statement 1 is insufficient

Statement 2

xy is odd

=> xy is odd if and only if x is odd and y is odd

=> x + y = odd + odd = even

Statement 2 is sufficient

Hence option B

bionication · Answer

emanresu1 , 'y' could be even OR odd for 'x' + '2y' to result in an odd integer. Therefore, statement (a) does not give us sufficient information as to whether integer 'y' is indeed an odd integer. Therefore, plugging it into the question stem (x+y) may result in an even OR an odd expression, hence insufficient. Hope that helps

Nikhil · Answer

If both x and y are odd then x + y becomes even. You only know the even/odd nature of x and y can be either even or odd. Hence statement 1 is insufficient

Example

x = 3 and y = 6 => x + y = 3 + 6 = 9 odd

x = 3 and y = 7 => x + y = 3 + 7 = 10 even

emanresu1 · Answer

THANK YOU! This Quant pressure has me flubbing even simple things like thinking 7+7 is odd (yes, I really thought that during my practice exam, and thus the ridiculous confusion)

Krabhay · Answer

x + y will be even only if either both of them are even or both of them are odd.
Statement 1: x + 2y is odd. 
2y is a multiple of 2 therefore it can’t be odd.
So, x is odd. But we don’t know if ‘y’ is odd or even.
Hence, Insufficient.
Statement 2: ‘xy’ is odd. This is only possible if ‘x’ and ‘y’ both are odd.
Since, ‘x’ and ‘y’ are odd, x + y is even. Hence, Sufficient.

meenakshimiyer · Answer

x + y will be even only if either both of them are even or both of them are odd.
Statement 1: x + 2y is odd. 
2y is a multiple of 2 therefore it can’t be odd.
So, x is odd. But we don’t know if ‘y’ is odd or even.
Hence, Insufficient.
Statement 2: ‘xy’ is odd. This is only possible if ‘x’ and ‘y’ both are odd.
Since, ‘x’ and ‘y’ are odd, x + y is even. Hence, Sufficient.

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If x and y are integers, is x + y even?

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