AS #54

[oquiella]

54. If xyz ≠ 0, is x3 + y3 + z3 = 0?

(1) x2 + y2 + z2 = xy + yz + xz
(2) x + y + z = 0

[Matt@VeritasPrep]

S1::

x² + y² + z² = xy + yz + xz

x² - xy + y² - yz + z² - xz = 0

subtract xy, yz, and zx from both sides:

x² - 2xy + y² - 2yz + z² - 2zx = -(xy + yz + zx)

add x², y², and z² to both sides:

2x² - 2xy + 2y² - 2yz + 2z² - 2zx = (x² + y² + z²) - (xy + yz + zx)

(x - y)² + (y - z)² + (z - x)² = (x² + y² + z²) - (xy + yz + zx)

But we know from S1 that the right hand side of this equation is 0! So

(x - y)² + (y - z)² + (z - x)² = 0

and (x - y) = 0, (y - z) = 0, and (z - x) = 0.

Hence x = y = z, so the question reduces to "Is 3x³ = 0?" Since x is nonzero, the answer is NO, and S1 is SUFFICIENT.


S2::

Suppose that x + y + z = 0 and x³ + y³ + z³ = 0. Since x = -(y + z) and x³ = -(y³ + z³), we have

-(y + z)³ = -(y³ + z³), or

-3y²z - 3yz² = 0, or

0 = y²z + yz², or

0 = yz(y + z)

Since y ≠ 0 and z ≠ 0 (from the prompt), we know that (y + z) = 0, and y = -z. But then x + y + z = 0 is really x + (-z) + z = 0, or x = 0.

Since this leads to a contradiction, we cannot have both (x + y + z) = 0 and (x³ + y³ + z³ = 0), so x³ + y³ + z³ ≠ 0, and this statement is SUFFICIENT.

But I need to put a stern warning in bold caps:

LIKE SOME OF YOUR OTHER QUESTIONS, THIS PROBLEM IS BROKEN: S1 and S2 GIVE ENTIRELY DIFFERENT ANSWERS WITH NO OVERLAP, SOMETHING THAT IS NEVER ALLOWED ON THE GMAT. (S1 says that y = z, but S2 says that y = -z, forcing z = 0 and contradicting the prompt.) THE SOURCE OF THESE QUESTIONS SHOULD BE TREATED WITH SKEPTICISM, AS THE QUESTION WRITERS DO NOT UNDERSTAND A FUNDAMENTAL ASPECT OF THE GMAT.

[oquiella]

S1::

x² + y² + z² = xy + yz + xz

x² - xy + y² - yz + z² - xz = 0

subtract xy, yz, and zx from both sides:

x² - 2xy + y² - 2yz + z² - 2zx = -(xy + yz + zx)

add x², y², and z² to both sides:

2x² - 2xy + 2y² - 2yz + 2z² - 2zx = (x² + y² + z²) - (xy + yz + zx)

(x - y)² + (y - z)² + (z - x)² = (x² + y² + z²) - (xy + yz + zx)

But we know from S1 that the right hand side of this equation is 0! So

(x - y)² + (y - z)² + (z - x)² = 0

and (x - y) = 0, (y - z) = 0, and (z - x) = 0.

Hence x = y = z, so the question reduces to "Is 3x³ = 0?" Since x is nonzero, the answer is NO, and S1 is SUFFICIENT.


S2::

Suppose that x + y + z = 0 and x³ + y³ + z³ = 0. Since x = -(y + z) and x³ = -(y³ + z³), we have

-(y + z)³ = -(y³ + z³), or

-3y²z - 3yz² = 0, or

0 = y²z + yz², or

0 = yz(y + z)

Since y ≠ 0 and z ≠ 0 (from the prompt), we know that (y + z) = 0, and y = -z. But then x + y + z = 0 is really x + (-z) + z = 0, or x = 0.

Since this leads to a contradiction, we cannot have both (x + y + z) = 0 and (x³ + y³ + z³ = 0), so x³ + y³ + z³ ≠ 0, and this statement is SUFFICIENT.

But I need to put a stern warning in bold caps:

LIKE SOME OF YOUR OTHER QUESTIONS, THIS PROBLEM IS BROKEN: S1 and S2 GIVE ENTIRELY DIFFERENT ANSWERS WITH NO OVERLAP, SOMETHING THAT IS NEVER ALLOWED ON THE GMAT. (S1 says that y = z, but S2 says that y = -z, forcing z = 0 and contradicting the prompt.) THE SOURCE OF THESE QUESTIONS SHOULD BE TREATED WITH SKEPTICISM, AS THE QUESTION WRITERS DO NOT UNDERSTAND A FUNDAMENTAL ASPECT OF THE GMAT.

Hi Matt,

I just started the DS questions from Aristotle DS boosters. and I'm finding it hard to grasp on a few due to having to test the answer in the inequalities then go back to the original and making sure it aligns. Should I discontinue using them? Are these unrealistic for the GMAT. I've been feeling like its been a little flawed, but everyone say its really good practice. What would you suggest? I have access to Veritas Quiz bank as well.

[Matt@VeritasPrep]

Hi Matt,

I just started the DS questions from Aristotle DS boosters. and I'm finding it hard to grasp on a few due to having to test the answer in the inequalities then go back to the original and making sure it aligns. Should I discontinue using them? Are these unrealistic for the GMAT. I've been feeling like its been a little flawed, but everyone say its really good practice. What would you suggest? I have access to Veritas Quiz bank as well.

Hi Oquiella!

I don't know anything about Aristotle -- I hadn't heard of them before they came up here -- so I can't say whether their materials are good or not, but if these questions are reflective of their math curriculum, I'm skeptical: it's troubling to have GMAT questions with flaws like these. Everybody makes mistakes, but there shouldn't be that many questions (even in a 200 page book) with any issues at all.

I'm not impartial, but I like the VP questions, especially since I wrote a few of them.