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

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Re: If x and y are different integers and x^2 = xy, which of the [#permalink]
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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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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) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

Let’s simplify the given equation:

x^2 = xy

x^2 - xy = 0

x(x - y) = 0

x = 0 or x - y = 0

x = 0 or x = y

Notice that it is given in the question that x and y are different integers; therefore x = y is not possible. Thus, it must be true that x = 0. None of the other Roman numerals need to be true, as we can see by letting x = 0 and y = 1.

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If x and y are different integers and x^2 = xy, which of the [#permalink]
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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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Re: If x and y are different integers and x^2 = xy, which of the [#permalink]
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
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Re: If x and y are different integers and x^2 = xy, which of the [#permalink]
Bunuel wrote:
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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x^2-xy=0
x(x-y)=0
Since x &y are different
x=0

IMO A

Posted from my mobile device
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Re: If x and y are different integers and x^2 = xy, which of the [#permalink]
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Synthesising the equation given in the question stem, $$x^2$$ = xy, we have,

$$x^2$$ – xy = 0

x(x-y) = 0.

This means x = 0 or x=y. But, question clearly says that x and y are different integers. Therefore, x=y is not true.

Therefore, x=0 is always true.
Statement I is always true. We can eliminate answer options B and C.

Since x≠y, y can never be 0. Therefore, statement II is always false. Answer option E can be eliminated.

If x = -y, x+y = 0. But, from the question data, we know that x – y = 0. Therefore, statement III is always false. Answer option D can be eliminated.

The correct answer option is A.

Hope that helps!
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