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# If x and y are integers and √x−√y equals an odd integer, which of the

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Joined: 02 Sep 2009
Posts: 46129
If x and y are integers and √x−√y equals an odd integer, which of the [#permalink]

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13 Apr 2016, 05:27
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Question Stats:

79% (01:09) correct 21% (01:11) wrong based on 107 sessions

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If x and y are integers and √x−√y equals an odd integer, which of the following must be an even integer?

I. xy
II. x^2 + y^2
III. x/y

A. I only
B. I and II
C. II​ only
D. I, II, and III
E. None of these

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Re: If x and y are integers and √x−√y equals an odd integer, which of the [#permalink]

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13 Apr 2016, 06:25
1
sqrt(x) - sqrt(y) = odd
sqrt(x) + sqrt(y) must also be = odd

(sqrt(x) - sqrt(y))*(sqrt(x) + sqrt(y)) = odd * odd
x - y = odd
even - odd = odd
odd - even = odd

I) xy = even? Yes. Since one term is even

II) x^2 + y^2 = even?
x^2 + y^2 - 2xy = odd
x^2 + y^2 = odd + even = odd
So x^2 + y^2 is not even

III) x/y = even?
If x = 6 and y = 3 then yes
If x = 3 and y = 2 then no
So x/y is not always even.

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If x and y are integers and √x−√y equals an odd integer, which of the [#permalink]

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14 Apr 2016, 00:01
If x and y are integers and √x−√y equals an odd integer, which of the following must be an even integer?

I. xy
II. x^2 + y^2
III. x/y

A. I only
B. I and II
C. II​ only
D. I, II, and III
E. None of these

Given that √x−√y = odd Int.

This means that one of the 2 should be odd and the other even. (Even/odd - Odd/Even = Odd). Implies x or y is even and the other is odd.

With this in mind:

I must be true - odd x even always even. (eliminate C and E).

II Not true. E + O is always odd. (Eliminate D and B).

Need not test the III.

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Re: If x and y are integers and √x−√y equals an odd integer, which of the [#permalink]

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14 Apr 2016, 12:30
Bunuel wrote:
If x and y are integers and √x−√y equals an odd integer, which of the following must be an even integer?

I. xy
II. x^2 + y^2
III. x/y

A. I only
B. I and II
C. II​ only
D. I, II, and III
E. None of these

$$\sqrt{x} - \sqrt{y}$$ = Odd number

So, X or Y can be Even / Odd; however both can not be even.

Check using properties of Number system

I. xy

Product of an Odd and Even Number must always be Even Number

II. x^2 + y^2

Square of an Odd number is Odd Number and Square of an even number is even Number
Further Odd + Even Number = Odd Number

III. $$\frac{x}{y}$$

Now, x / y can be even / odd

If x is even and y odd then $$\frac{x}{y}$$ is even

However if x is odd and y even then $$\frac{x}{y}$$ is odd

Thus using property of Number system we can find that option (A) is correct !!
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Abhishek....

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Joined: 12 Aug 2015
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GRE 1: 323 Q169 V154
Re: If x and y are integers and √x−√y equals an odd integer, which of the [#permalink]

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23 Nov 2016, 01:39
Exponent does not affect the even/odd nature of x,y

Hence x-y=odd
so xy is even
x^2+y^2 is odd
And x/y may or may not be an integer

So A
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Re: If x and y are integers and √x−√y equals an odd integer, which of the [#permalink]

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17 Apr 2017, 06:02
Option A

x & y are integers such that $$\sqrt{x} - \sqrt{y}$$ = Odd Integer
Hence, x & y must perfect squares AND $$\sqrt{x}$$ & $$\sqrt{y}$$ must be either Odd & Even or Even & Odd.

I. xy = E
II. x^2 + y^2 = E^2 + O^2 or O^2 + E^2 = E + O or O + E = O
III. x/y = E/O or O/E = E or O or F
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Re: If x and y are integers and √x−√y equals an odd integer, which of the   [#permalink] 17 Apr 2017, 06:02
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