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Math Expert V
Joined: 02 Sep 2009
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10 00:00

Difficulty:   95% (hard)

Question Stats: 40% (01:45) correct 60% (02:11) wrong based on 223 sessions

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If $$x$$ and $$y$$ are integers, is $$y$$ an even integer?

(1) $$4y^2+3x^2=x^4+y^4$$

(2) $$y=4-x^2$$

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Math Expert V
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Official Solution:

(1) $$4y^2+3x^2=x^4+y^4$$

Rearrange: $$3x^2-x^4=y^4-4y^2$$;

$$x^2(3-x^2)=y^2(y^2-4)$$.

Notice that LHS is even for any value of $$x$$: if $$x$$ is odd then $$3-x^2=odd-odd=even$$ and if $$x$$ is even, then the product is naturally even. So, $$y^2(y^2-4y)$$ is also even, but in order for it to be even $$y$$ must be even, since if $$y$$ is odd, then $$y^2(y^2-4y)=odd*(odd-even)=odd*odd=odd$$. Sufficient.

(2) $$y=4-x^2$$

If $$x=odd$$ then $$y=even-odd=odd$$ but if $$x=even$$ then $$y=even-even=even$$. Not sufficient.

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wow this was cool. pluging in 1, 0, 2 helps
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Bunuel wrote:
Official Solution:

(1) $$4y^2+3x^2=x^4+y^4$$

Rearrange: $$3x^2-x^4=y^4-4y^2$$;

$$x^2(3-x^2)=y^2(y^2-4)$$.

Notice that LHS is even for any value of $$x$$: if $$x$$ is odd then $$3-x^2=odd-odd=even$$ and if $$x$$ is even, then the product is naturally even. So, $$y^2(y^2-4y)$$ is also even, but in order for it to be even $$y$$ must be even, since if $$y$$ is odd, then $$y^2(y^2-4y)=odd*(odd-even)=odd*odd=odd$$. Sufficient.

(2) $$y=4-x^2$$

If $$x=odd$$ then $$y=even-odd=odd$$ but if $$x=even$$ then $$y=even-even=even$$. Not sufficient.

I did it without regrouping.

1) 4y^2 - is alway even; 3x^2 could be even or odd, depending on X;
x^4 will be even or odd, depending on X, but it will be the same as above; y^4 is unknown?
e + e = e => e = e + something, then this something is even
e + o = o => o = o + something, then this something is even

2) Not sufficient
Intern  B
Joined: 10 May 2017
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Hello Sir,

I assumed s1 in this way, kindly let me know if my assumption is correct.

After re-grouping to X on one side and Y on the other,

I found that no matter if X is odd or even, the solution of X^2(3-X^2) is always even. Therefore inorder to achieve this even number, Y must always be even since Odd does not satisfy the X^2(3-X^2).

Thank you Sir.
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Why am I not able to see the answer choices for Quant questions?
Math Expert V
Joined: 02 Sep 2009
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katkepin wrote:
Why am I not able to see the answer choices for Quant questions?

This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

I suggest you to go through the following posts:
ALL YOU NEED FOR QUANT.

Hope this helps.
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this was very helpful! thank you
Bunuel wrote:
Hope this helps. Re: M27-02   [#permalink] 11 Sep 2018, 12:54
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