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# If x and y are integers, is the sum of x and y even?

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If x and y are integers, is the sum of x and y even?  [#permalink]

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11 Apr 2016, 04:07
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69% (01:12) correct 31% (01:20) wrong based on 92 sessions

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If x and y are integers, is the sum of x and y even?

(1) $$x^2$$ = $$y^2$$
(2) $$x^3$$ = $$y^3$$

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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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11 Apr 2016, 11:25
find out x+y= even or not.
1. x^2=y^2 , by rearranging, we can write this as (x+y)(x-y)=0; x+y=0 and 0 is even number so sufficient.
2. x^3-y^3 =0 by rearranging we do not about x+y so not sufficient.

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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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Updated on: 14 Apr 2016, 21:14
Bunuel wrote:
If x and y are integers, is the sum of x and y even?

(1) x^2 = y^2
(2) x^3 = y^3

x + y = even?

Statement 1: x^2 = y^2
Case 1:
Either x = y (x = 2 and y = 2)
or x = -y (x = 2 and y = -2)
Hence x + y = 2x or 0
Yes, the sum is even
Sufficient

Statement 2: x^3 = y^3
x = y (x = 2 and y = 2 OR x = -2 and y = -2)
Therefore x + y = 2x
Yes, the sum is even.
Sufficient

Option D

Originally posted by OptimusPrepJanielle on 14 Apr 2016, 02:09.
Last edited by OptimusPrepJanielle on 14 Apr 2016, 21:14, edited 1 time in total.
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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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14 Apr 2016, 02:12
lipsi18 wrote:
find out x+y= even or not.
1. x^2=y^2 , by rearranging, we can write this as (x+y)(x-y)=0; x+y=0 and 0 is even number so sufficient.
2. x^3-y^3 =0 by rearranging we do not about x+y so not sufficient.

Hi lipsi18,

In the statement 1, either of (x+y) or (x-y) can be zero.
Hence we cannot say that x + y = 0
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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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14 Apr 2016, 09:06
Is it B ?
If so, will provide my line of reasoning.
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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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14 Apr 2016, 21:13
subhamoycomputer wrote:
Is it B ?
If so, will provide my line of reasoning.

Hi subhamoycomputer,

The correct option here should be D.
However do post your reasoning as to why you thing option A is insufficient.
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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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14 Apr 2016, 21:25
If x and y are integers, is the sum of x and y even?

(1) x^2 = y^2
(2) x^3 = y^3

x + y even?

Statement 1:

x^2 = y^2. This implies x = y or x =-y.

Either way x+y will be even (0 or 2x).

Statement 2:

X^3 = y^3.

Implies x = y.

So x + y is 2x, which is even.

So each statement is individually sufficient.

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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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15 Apr 2016, 05:03
Here in both the cases we can say that both x and y must have the same odd/even nature
hence x+y=even
hence D
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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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16 Apr 2016, 02:43
If x and y are integers, is the sum of x and y even?

(1) x^2 = y^2
(2) x^3 = y^3

There are 2 variables (x and y) in the original condition. In order to match the number of variables to the number of equations, we need 2 more equations. Since the condition 1) and the condition 2) each has 1 equation, there is high chance that C is the correct answer. Using both the condition 1) and the condition 2), we get 1)=2), from which we can obtain x=y=odd or x=y=even. The answer is always yes and the conditions are sufficient. Therefore, the correct answer is D.

For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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18 Apr 2016, 03:31
Bunuel wrote:
If x and y are integers, is the sum of x and y even?

(1) x^2 = y^2
(2) x^3 = y^3

The Answer is D..Check for pairs (0,0) ...(1,1)....(-1,-1)
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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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18 Apr 2016, 03:44
1
Balajikarthick1990 wrote:
Bunuel wrote:
If x and y are integers, is the sum of x and y even?

(1) x^2 = y^2
(2) x^3 = y^3

The Answer is D..Check for pairs (0,0) ...(1,1)....(-1,-1)

You are correct...
But for (1), check for |x|=|y|.. May be important in some other Q
x=1, y=-1 ; x=-1, y=1; x=1,y=1; x=-1, y=-1
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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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18 Apr 2016, 18:29
Let me understand, is 0 considered even?

Because if we use x = -y
Then the answer would be 0 for option A
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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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18 Apr 2016, 20:07
douglasvg wrote:
Let me understand, is 0 considered even?

Because if we use x = -y
Then the answer would be 0 for option A

hi,
0 is EVEN but it is neither NEGATIVE nor POSITIVE...
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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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20 Dec 2018, 07:59

Q is asking: Are both x and y even or odd?

For St 1, I took |x| = |y|
which is same as:
x = y and x = -y
or
x-y = 0
or
x + y = 0

Hence implying, both x and y have to be even/odd.

What is St 2 saying? How do I interpret odd powers of integer?
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Re: If x and y are integers, is the sum of x and y even?  [#permalink]

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20 Dec 2018, 08:20
1) x²=y²
So, x²-y²=0
So, (x+y)(x-y)=0
Either x+y=0 where 0 is even.
Or x-y=0 where 0 is even. If the difference of two integers is even, their sum is also even.

So statement (1) is sufficient.

(2) Cube of a number preserves the sign.
So x³=y³ means x=y, say x=y=a
So the sum will always be even =2a.

So statement (2) is sufficient.

Both are statements alone are sufficient so option D is the correct answer

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If x and y are integers, is the sum of x and y even?  [#permalink]

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20 Dec 2018, 08:24
if $$x^ a = y ^a$$, where a is an even integer
we get x = +/- y

But, if $$x^ b = y ^b$$, where b is an odd integer
we get x = y

Good news, odd power are even easier to handle

Hence for Statement 2, sum of x and y = 2x or 2y = even

Q is asking: Are both x and y even or odd?

For St 1, I took |x| = |y|
which is same as:
x = y and x = -y
or
x-y = 0
or
x + y = 0

Hence implying, both x and y have to be even/odd.

What is St 2 saying? How do I interpret odd powers of integer?

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If x and y are integers, is the sum of x and y even?   [#permalink] 20 Dec 2018, 08:24
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