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S98-14

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S98-14  [#permalink]

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New post 16 Sep 2014, 01:52
1
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

74% (00:58) correct 26% (00:43) wrong based on 31 sessions

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\(x^6 - y^6 =\)

A. \((x^3 + y^3)(x^2 + y^2)(x - y)\)
B. \((x^3 - y^3)(x^3 - y^3)\)
C. \((x^2 + y^2)(x^2 + y^2)(x + y)(x - y)\)
D. \((x^4 + y^4)(x + y)(x - y)\)
E. \((x^3 + y^3)(x^2 + xy + y^2)(x - y)\)

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Re S98-14  [#permalink]

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New post 16 Sep 2014, 01:52
Official Solution:

\(x^6 - y^6 =\)

A. \((x^3 + y^3)(x^2 + y^2)(x - y)\)
B. \((x^3 - y^3)(x^3 - y^3)\)
C. \((x^2 + y^2)(x^2 + y^2)(x + y)(x - y)\)
D. \((x^4 + y^4)(x + y)(x - y)\)
E. \((x^3 + y^3)(x^2 + xy + y^2)(x - y)\)


An efficient way to attack this problem is to rephrase the expression in the stem. Since a sixth power is a square, we can look at the difference of sixth powers as the difference of squares, and factor:
\(x^6 - y^6 = (x^3)2 - (y^3)2 = (x^3 + y^3)(x^3 - y^3)\)

This new expression is not among the answer choices, but as a search strategy, we should focus on these factors.

Choice (A) contains \((x^3 + y^3)\), so we should determine whether the rest of (A) works out too \((x^3 - y^3)\). The terms \((x^2 + y^2)(x - y)\) look promising, since they give us \(+x^3\) and \(-y^3\), but we get cross-terms that don't cancel: \((x^2 + y^2)(x - y) = x^3 - x^2y + xy^2 - y^3\).

Choice (B) is close but not right. \((x^3 - y^3)(x^3 - y^3) = (x^3 - y^3)^2\), which also gives us cross-terms that don't cancel: \((x3 - y3)^2 = x^6 - 2x^3y^3 + y^6\). (Moreover, the sign is wrong on the \(y^6\) term.)

Let's skip to choice (E), since we see \((x^3 + y^3)\). The remaining terms multiply out as follows:

\((x^2 + xy + y^2)(x - y) = x^3 + x^2y + xy^2 - x^2y - xy^2 - y^3 = x^3 - y^3\). This is what we were looking for. We can verify that the other answer choices do not multiply out to \(x^6 - y^6\) exactly.

The correct answer is (E).

Of course, you can also plug simple numbers. Don't pick 0 for either \(x\) or \(y\), because too many terms will cancel. Also, don't pick the same number for \(x\) and \(y\), because then the result is 0 (and every answer choice gives you 0 as well). But if you pick 2 and 1, you can keep the quantities relatively small and still eliminate wrong answers.

\(2^6 - 1^6 = 64 - 1 = 63\). Our target is 63.
\((8 + 1)(4 + 1)(2 - 1) = 45\)
\((8 - 1)(8 - 1) = 49\)
\((4 + 1)(4 + 1)(2 + 1)(2 - 1) = 75\)
\((16 + 1)(2 + 1)(2 - 1) = 51\)
\((8 + 1)(4 + 2 + 1)(2 - 1) = 63\)


Answer: E
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DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: S98-14  [#permalink]

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New post 02 Jul 2018, 01:48
Any further questions like the above?
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Re: S98-14  [#permalink]

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New post 02 Jul 2018, 02:32
Madhavi1990 wrote:
Any further questions like the above?


Plenty.

7. Algebra



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New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: S98-14 &nbs [#permalink] 02 Jul 2018, 02:32
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