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# If x and y are different integers and x^2 = xy, which of the

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Math Expert
Joined: 02 Sep 2009
Posts: 47166
If x and y are different integers and x^2 = xy, which of the  [#permalink]

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21 Feb 2014, 01:00
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The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x and y are different integers and x^2 = xy, which of the following must be true?

I. x = 0
II. y = 0
III. x = -y

(A) l only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

Problem Solving
Question: 106
Category: Arithmetic; Algebra Operations on rational numbers; Second-degree equations
Page: 75
Difficulty: 600

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Posts: 47166
Re: If x and y are different integers and x^2 = xy, which of the  [#permalink]

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21 Feb 2014, 01:01
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SOLUTION

If x and y are different integers and x^2 = xy, which of the following must be true?

I. x = 0
II. y = 0
III. x = -y

(A) l only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

$$x^2=xy$$ --> $$x(x-y)=0$$ --> either $$x=0$$ or $$x=y$$ but as given that $$x$$ and $$y$$ are different numbers than the second option is out and we have: $$x=0$$. So only I is always true (in fact because of the same reason that $$x$$ and $$y$$ are different numbers II and III are never true).

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Re: If x and y are different integers and x^2 = xy, which of the  [#permalink]

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21 Feb 2014, 02:18
1
Given Facts -
1. x and y are different integers
2. x^2 = xy

I - x=0
then x^2 = 0
and xy = 0 , for any value of y
thus x^2 = xy - True for any value of y

II - y = 0
if x=1 then x^2 = 1
and xy = 0
Thus x^2 not equal to xy and thus not true for any value of x

III - x = -y
x^2 = (-y)^2 = y^2
xy = (-y)*(y)=-y^2
Thus x^2 not equal to xy and thus not true for any value of x,y

Hence since I only is true for all values Option (A) is the correct answer.
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Re: If x and y are different integers and x^2 = xy, which of the  [#permalink]

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21 Feb 2014, 09:35
1
If x and y are different integers and x^2 = xy, which of the following must be true?

I. x = 0
II. y = 0
III. x = -y

Sol.: From the question stem,
x^2 – xy = 0
x ( x – y) = 0
⇨ Either x = 0 or x = y
Since x and y are two different integers, => x= 0.
Hence (A)
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Joined: 09 Mar 2016
Posts: 658
Re: If x and y are different integers and x^2 = xy, which of the  [#permalink]

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09 Apr 2018, 08:13
Bunuel wrote:
SOLUTION

If x and y are different integers and x^2 = xy, which of the following must be true?

I. x = 0
II. y = 0
III. x = -y

(A) l only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

$$x^2=xy$$ --> $$x(x-y)=0$$ --> either $$x=0$$ or $$x=y$$ but as given that $$x$$ and $$y$$ are different numbers than the second option is out and we have: $$x=0$$. So only I is always true (in fact because of the same reason that $$x$$ and $$y$$ are different numbers II and III are never true).

Bunuel why this option is not valid III. x = -y

if y = -2
and x= y ---> x = -2

then $$-2^2 = (-2*)(-2)$$ --> $$4 =4$$

can you explain ? pleaese
Math Expert
Joined: 02 Sep 2009
Posts: 47166
If x and y are different integers and x^2 = xy, which of the  [#permalink]

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09 Apr 2018, 08:26
dave13 wrote:
Bunuel wrote:
SOLUTION

If x and y are different integers and x^2 = xy, which of the following must be true?

I. x = 0
II. y = 0
III. x = -y

(A) l only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

$$x^2=xy$$ --> $$x(x-y)=0$$ --> either $$x=0$$ or $$x=y$$ but as given that $$x$$ and $$y$$ are different numbers than the second option is out and we have: $$x=0$$. So only I is always true (in fact because of the same reason that $$x$$ and $$y$$ are different numbers II and III are never true).

Bunuel why this option is not valid III. x = -y

if y = -2
and x= y ---> x = -2

then $$-2^2 = (-2*)(-2)$$ --> $$4 =4$$

can you explain ? pleaese

This does not make sense.

If y = -2 and x = -y, then x = -(-2) = 2. What x= y has to do with this? x = y is not possible at all because the stem says that x and y are different integers...

Also, the question asks which of the following MUST be true not COULD be true. Even if there would exist x and y, for which x = -y, would satisfy the stem, still this would not be sufficient to say that III MUST be true. MUST be true means true for ALL possible values, not for some specific values only.

Finally, you should be careful with brackets. It's math, they DO matter. -2^2 means -(2^2) = -4, while (-2)^2 = 4.
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If x and y are different integers and x^2 = xy, which of the &nbs [#permalink] 09 Apr 2018, 08:26
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