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# M26-36

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Math Expert
Joined: 02 Sep 2009
Posts: 47184

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16 Sep 2014, 01:26
00:00

Difficulty:

65% (hard)

Question Stats:

42% (01:03) correct 58% (01:32) wrong based on 33 sessions

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If $$x$$, $$y$$ and $$z$$ are non-zero numbers, what is the value of $$\frac{x^3+y^3+z^3}{xyz}$$?

(1) $$xyz=-6$$

(2) $$x+y+z=0$$

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Math Expert
Joined: 02 Sep 2009
Posts: 47184

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16 Sep 2014, 01:26
1
Official Solution:

(1) $$xyz=-6$$. Infinitely many combinations of $$x$$, $$y$$ and $$z$$ are possible which will give different values of the expression in the stem: try $$x=y=1$$ and $$y=-6$$ or $$x=1$$, $$y=2$$, $$z=-3$$. Not sufficient.

(2) $$x+y+z=0$$. Rearrange: $$x=-(y+z)$$ and substitute this value of $$x$$ into the expression in the stem to get $$\frac{x^3+y^3+z^3}{xyz}=\frac{-(y+z)^3+y^3+z^3}{xyz}=\frac{-y^3-3y^2z-3yz^2-z^3+y^3+z^3}{xyz}=\frac{-3y^2z-3yz^2}{xyz}=\frac{-3yz(y+z)}{xyz}$$, as $$x=-(y+z)$$ then:

$$\frac{-3yz(y+z)}{xyz}=\frac{-3yz*(-x)}{xyz}=\frac{3xyz}{xyz}=3$$. Sufficient.

Must know for the GMAT:$$(x+y)^3=(x+y)(x^2+2xy+y^2)=x^3+3x^2y+3xy^2+y^3$$ and $$(x-y)^3=(x-y)(x^2-2xy+y^2)=x^3-3x^2y+3xy^2-y^3$$.

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Intern
Joined: 16 Jul 2014
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04 Nov 2014, 18:21
Once it gets to this stage:

−3yz(y+z)
xyz

yz can be canceled out so we're only left with:

−3(y+z)
x

And since x = -(y+z)

3x
x

= 3
Manager
Joined: 17 May 2017
Posts: 144
GPA: 3

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30 Aug 2017, 01:43
Can I expect such type of questions in GMAT
this is more or less like CAT(Entrance Exam for Indian Institute of Management) Question
Manager
Joined: 20 Feb 2017
Posts: 119
Location: India
Concentration: Operations, Strategy
WE: Engineering (Other)

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17 Dec 2017, 13:29
THERE IS A SIMPLE PROPERTY
IF X+Y+Z = 0
THEN X^3+Y^3+Z^3 = 3XYZ
HOPE IT HELPS
_________________

If you feel the post helped you then do send me the kudos (damn theya re more valuable than \$)

Manager
Joined: 30 May 2017
Posts: 118
Location: United States
Schools: HBS '21
GMAT 1: 690 Q50 V32
GRE 1: Q168 V164
GPA: 3.57

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05 May 2018, 18:46
THERE IS A SIMPLE PROPERTY
IF X+Y+Z = 0
THEN X^3+Y^3+Z^3 = 3XYZ
HOPE IT HELPS

I assume we only expect the property above when NONE of X,Y and Z is equal to zero.
_________________

Kindly press the +1Kudos if you like the explanation. Thanks a lot!!!

M26-36 &nbs [#permalink] 05 May 2018, 18:46
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# M26-36

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