If x and y are positive integers, is x^2 – y^2 an even integer?

Question

(1) 3 is a factor of x + y.
(2) x – y is an odd integer.

chetan2u · Answer

Hi,
x^2 - y^2 will be EVEN only if both x and y are same type of integers, either BOTH odd or BOTH even..
Also x^2 - y^2 = (x-y)(x+y)... so either of x-y or x+y is even
NOTE- if x-y is even, x+y will also be even OR both will be ODD..

lets see the statements--

(1) 3 is a factor of x + y.
we do not know if x+y has a even number factor 
say x= 6 and y = 9... ans is NO
if x= 6 and y=12... ans is YES
Insuff

(2) x – y is an odd integer.
If x-y is ODD, x+y will also be ODD, 
so our answer is NO
Suff

B

iliavko · Answer

Are you sure this is correct?..

(x+y)(x-y)= even if :

even * even = even
odd * even = even
even * odd = even

so from 1) no relevant info.

from 2) we get that x-y = odd this is only possible if:

odd - even = odd
even - odd = odd

so we get no info on x,y being odd or even.

So how can we say by using statement 2) if (x+y)(x-y) is even?

We know that (x-y) is odd, but the left (x+y) can be odd or even and we can't extract any "internal info" on x or y from inside the parenthesis.

(x+y)(x-y) is even? -> it's a question, not a statement, so we can't use it as base information to figure out if (x+y) is even or odd even knowing that (x-y) is odd, so we get E.

Am I missing something?

Bunuel , can you be so kind to take a look at this?

Thank you.

Bunuel · Answer

You are saying yourself that x - y to be odd x must be odd and y even OR x must be even and y odd. What would be x + y then?

x = odd and y = even --> x + y= odd + even = odd;
x = even and y = odd --> x + y= even + odd = odd.

Thus in any case x + y = odd.

iliavko · Answer

Thank you, Bunuel , got it. I need to get some rest for today

These odd\even questions fry my brain!

hsbinfy · Answer

answer is B

lets take 2nd statement

there are only two cases here:
E-O=O, so E+O=O
O-E=O , so O+E =O

so the expression is always odd.hence our answer

for 1st case
x+y is factor of 3 so x+y can be even or odd.not suff

Divyadisha · Answer

In what case can x^2 – y^2 be even?

Both x and y are even. then E^2-E^2= Even
or Both x and y are odd. Then O^2 - O^2= Even.

(1) 3 is a factor of x + y.
Let's say x+y = 3, where x= 2 and y = 1
In this case x^2 – y^2 is not even

If x+y= 6, where x=4 and y=2

In this case x^2 – y^2 is even

Not sufficient.

(2) x – y is an odd integer.

For x-y to be odd. either x or y is even and the other integer is odd.

In both the cases, square of one integer will be odd and the other integer will be even. And O-E or E-O will be odd.

Sufficient.

B is the answer

stonecold · Answer

Great Questions to clear the concepts on evens and odds
Given info=>
x and y are integers (this is a very important piece of info)

And we need to see whether x^2-y^2 is even or not .

Lets look at statements here
Statement 1
3 is a factor of x+y 
This statement can be proven insufficient by taking test cases pretty easily
let x+y = 3
hence x-y will be odd too
so x^y-y^2 => (x+y)*(x-y) will be odd*odd => odd
Next lets take x+y=6
so x-y will be even too
so x^y-y^2 => (x+y)*(x-y) will be even*even => even

hence Insufficient

Statement 2
Here we are given that x-y is odd
so for x-y to be odd , both x and y must have different even-odd nature
Hence x+y will be odd too.
Thus x^y-y^2 => (x+y)*(x-y) will be odd*odd => odd
Hence Sufficient

Hence B

For Quick Calculation we can memorise that 
x+y and x-y will always have the same even/odd nature

bumpbot · Answer

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If x and y are positive integers, is x^2 – y^2 an even integer?

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If x and y are positive integers, is x^2 – y^2 an even integer?

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