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# If x, y and z are three different non-negative integers, which of the

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Joined: 12 Sep 2015
Posts: 4064
If x, y and z are three different non-negative integers, which of the  [#permalink]

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17 Jun 2019, 09:25
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95% (hard)

Question Stats:

25% (02:42) correct 75% (02:23) wrong based on 80 sessions

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If x, y and z are three different non-negative integers, which of the following COULD be true?

i) $$|x-y|=|x+y|=|y-z|$$

ii) $$x^y = y^z$$

iii) $$x^3 + y^3 = z^3$$

A) i only
B) ii only
C) iii only
D) i and ii
E) i and iii

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If x, y and z are three different non-negative integers, which of the  [#permalink]

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Updated on: 17 Jun 2019, 12:11
given that x, y and z are three different non-negative integers so they can be any integer value =>0
i x=0,y=1,z=2
we get 1=1=1
and
$$x^y = y^z$$
we can take x=1,y=2 and z=0
sufficient
IMO D

GMATPrepNow wrote:
If x, y and z are three different non-negative integers, which of the following COULD be true?

i) $$|x-y|=|x+y|=|y-z|$$

ii) $$x^y = y^z$$

iii) $$x^3 + y^3 = z^3$$

A) i only
B) ii only
C) iii only
D) i and ii
E) i and iii

Originally posted by Archit3110 on 17 Jun 2019, 09:42.
Last edited by Archit3110 on 17 Jun 2019, 12:11, edited 2 times in total.
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Joined: 12 Sep 2015
Posts: 4064
Re: If x, y and z are three different non-negative integers, which of the  [#permalink]

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17 Jun 2019, 11:19
Top Contributor
Archit3110 wrote:
given that x, y and z are three different non-negative integers so they can be any integer value =>0
IMO A is sufficient only at x=0,y=1,x=2
we get 1=1=1
IMO A

Sorry, but the answer isn't A.
Keep trying!!
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Joined: 19 Oct 2018
Posts: 1077
Location: India
If x, y and z are three different non-negative integers, which of the  [#permalink]

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Updated on: 17 Jun 2019, 12:25
1
i) If x>y
x-y=x+y
y=0
|x|=|x|=|-z|
As z can't be negative, it must be equal to x
Not possible under given constraints

If x<y
y-x=x+y
x=0
Hence
|-y|=|y|=|y-z|
|y|=|y-z|
There are 2 cases again
If y>z
y=y-z
z=0(not possible)

If z>y
y=z-y
z=2y(possible)

ii)$$x^y=y^z$$
Put x=1, y=2 and z=0
$$(1)^2=2^0$$
1=1
Can be true

iii)
$$x^3+y^3=z^3$$
Fermat's last theorem, there are no three positive integer x,y and z satisfies $$x^n+y^n=z^n$$ for n>2
If any of the number is equal to 0, then the absolute value of other 2 numbers is equal which is against out constraints.
Not Possible!!

GMATPrepNow wrote:
If x, y and z are three different non-negative integers, which of the following COULD be true?

i) $$|x-y|=|x+y|=|y-z|$$

ii) $$x^y = y^z$$

iii) $$x^3 + y^3 = z^3$$

A) i only
B) ii only
C) iii only
D) i and ii
E) i and iii

Originally posted by nick1816 on 17 Jun 2019, 12:05.
Last edited by nick1816 on 17 Jun 2019, 12:25, edited 2 times in total.
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4064
Re: If x, y and z are three different non-negative integers, which of the  [#permalink]

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17 Jun 2019, 12:17
Top Contributor
GMATPrepNow wrote:
If x, y and z are three different non-negative integers, which of the following COULD be true?

i) $$|x-y|=|x+y|=|y-z|$$

ii) $$x^y = y^z$$

iii) $$x^3 + y^3 = z^3$$

A) i only
B) ii only
C) iii only
D) i and ii
E) i and iii

I created this question to illustrate the importance of checking the answer choices each time you analyze one of the statements. Here’s why:

Once we know that statement i COULD be true (it's true when x = 0, y = 1 and z = 2), we should check the answer choices….
ELIMINATE answer choices B and C, since they incorrectly state that statement i CANNOT be true.

Once we know that statement ii COULD be true (it's true when x = 8, y = 2 and z = 6), check the remaining answer choices….
ELIMINATE answer choices A and E, since they incorrectly state that statement ii CANNOT be true.

So, by the process of elimination, we know that the correct answer is D
Best of all, we’re able to determine the correct answer WITHOUT analyzing statement iii, which is great since statement iii is VERY TRICKY.

By the way, if you had a hard time determining whether statement iii could be true, you’re not alone!
Statement iii is something mathematicians have struggled with since 1637, when a French mathematician named Pierre de Fermat (shown here)…

…suggested that the equation $$a^3 + b^3 = c^3$$ cannot have a solution in which the values of a, b and c are POSITIVE integers.
In fact, Pierre de Fermat went even further to say that the equation $$a^n + b^n = c^n$$ cannot have a solution in which the values of a, b and c are POSITIVE integers, for n > 2.

Unfortunately Fermat never proved his theorem. In fact, the theorem (famously known as Fermat’s Last Theorem) was found after his death. Fermat had written the theorem in the margin of one of his textbooks. In addition to the theorem, Fermat wrote that he had a marvelous proof for it, but that it was too long to fit in the margin of the book.

Since then, pretty much every mathematician has tried to prove (or disprove) Fermat’s Last Theorem, but it took until 1994 (357 years later!!!) when it was successfully proved by a mathematician named Andrew Wiles.

Cheers,
Brent
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Re: If x, y and z are three different non-negative integers, which of the   [#permalink] 17 Jun 2019, 12:17
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