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Senior Manager  Joined: 23 Apr 2015
Posts: 283
Location: United States
Concentration: General Management, International Business
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For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 63% (02:13) correct 37% (02:36) wrong based on 145 sessions

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For Consecutive integers x, y and z, where x > y > z, which of the following CANNOT be the value of $$(x^2 - y^2)*(y^2 - z^2)$$ ?

A) 63
B) 99
C) 195
D) 276
E) 323

Originally posted by Senthil1981 on 11 Sep 2016, 20:10.
Last edited by Bunuel on 12 Sep 2016, 23:50, edited 2 times in total.
Renamed the topic and edited the question.
Senior Manager  Joined: 23 Apr 2015
Posts: 283
Location: United States
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Re: For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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[quote="accountrail"]Plz let me know,method to solve these kind of questions

The question mentions that the numbers are consecutive and $$x >y > z$$
so first simplify $$(x^2−y^2)∗(y^2−z^2) = (x - y) * (x + y) * (y - z) * (y + z)$$ and since $$x > y > z , x - y = 1$$ and $$y - z = 1$$
So what remains is $$(x +y ) * (y + z)$$ and since the numbers are consecutive, $$x +y$$ and $$y + z$$ has to odd, so only option which is even is D CANNOT be value.
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Re: For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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Senthil1981 wrote:
accountrail wrote:
Plz let me know,method to solve these kind of questions

The question mentions that the numbers are consecutive and $$x >y > z$$
so first simplify $$(x^2−y^2)∗(y^2−z^2) = (x - y) * (x + y) * (y - z) * (y + z)$$ and since $$x > y > z , x - y = 1$$ and $$y - z = 1$$
So what remains is $$(x +y ) * (y + z)$$ and since the numbers are consecutive, $$x +y$$ and $$y + z$$ has to odd, so only option which is even is D CANNOT be value.

This is too simple....even n odd
Say if I gave you D as 209.. and A as 35 and rest all same
What will be the answer
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Re: For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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2
OE:

Perhaps we should list a few of our squares here: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169. OK, that should take us far enough. Now, since we just need the differences between consecutive squares, we can just consider the intervals between values listed above: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25. Upon noticing that these intervals are only odd, one must conclude that a product of two of them must be odd and the answer is clearly D.
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Re: For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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Nevernevergiveup wrote:
OE:

Perhaps we should list a few of our squares here: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169. OK, that should take us far enough. Now, since we just need the differences between consecutive squares, we can just consider the intervals between values listed above: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25. Upon noticing that these intervals are only odd, one must conclude that a product of two of them must be odd and the answer is clearly D.

The main logic to this Q is that the answer has to be a multiple of CONSECUTIVE odd integers....
Incase no choice was EVEN, we would be looking into this aspect
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For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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Senthil1981 wrote:
accountrail wrote:
Plz let me know,method to solve these kind of questions

The question mentions that the numbers are consecutive and $$x >y > z$$
so first simplify $$(x^2−y^2)∗(y^2−z^2) = (x - y) * (x + y) * (y - z) * (y + z)$$ and since $$x > y > z , x - y = 1$$ and $$y - z = 1$$
So what remains is $$(x +y ) * (y + z)$$ and since the numbers are consecutive, $$x +y$$ and $$y + z$$ has to odd, so only option which is even is D CANNOT be value.

Also we can derive whole wuqation $$(x +y ) * (y + z)$$ in terms of x as $$(2x-1)(2x-3)$$ since $$y=x-1$$ and $$z=y-1=x-2$$

we can easily note that (2n+k) where k is any odd number indicates an odd number.
Hence product of two odd numbers is an odd number.
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Re: For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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Senthil1981 wrote:
For Consecutive integers x, y and z, where x > y > z, which of the following CANNOT be the value of $$(x^2 - y^2)*(y^2 - z^2)$$ ?

A) 63
B) 99
C) 195
D) 276
E) 323

2 ways:
1) x-odd y-even z-odd $$(odd^2-even^2)^(even^2-odd^2) = even$$
2) x-even y-odd z-even $$(even^2-odd^2)^(odd^2-even^2) =even$$

The only even answer we have is D) 276
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Re: For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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Continuing from Senthil1981's solution, (x+y)(y+z) can be represented in terms of y as (y+1+y)(y+y-1)--->(2y+1)(2y-1)--->4y^2-1
Now equate 4y^2-1 to each answer option and check if you are getting a valid integer value for y. It will work for all options excluding 276.Hence, D is the answer.
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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1
Hi All,

GMAT questions are almost always built around patterns - even if you don't realize that the pattern is there, you can probably do a bit of 'brute force' work and define the pattern. By extension, if you know the pattern, then you should be able to use that knowledge to your advantage to either answer the question immediately (or do another step or two of work to get the answer).

Here, we're given some specific facts to work with:
1) X, Y and Z are CONSECUTIVE integers
2) X > Y > Z

We're asked for what CANNOT be the value of (X^2 - Y^2)(Y^2 - Z^2).

Let's TEST VALUES and see if a pattern emerges...

IF... X = 3, Y = 2, Z = 1....
(9 - 4)(4 - 1) = (5)(3) = 15

So "15" is a possible answer. Also note that we ended up multiplying two ODD numbers together... Let's try another TEST....

IF... X = 4, Y = 3, Z = 2....
(16 - 9)(9 - 4) = (7)(5) = 35

So "35" is a possible answer. Notice that we again ended up multiplying two ODD numbers together... That looks like a pattern. If the end result is just going to be an ODD number every time, then there's clearly an answer that CANNOT be the value...

If you're not convinced yet, then try another example (and feel free to try as many as you like - as the numbers increase, you'll eventually hit all 4 of the possible answers, at which point you'll know which answer is NOT possible.

GMAT assassins aren't born, they're made,
Rich
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Re: For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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EMPOWERgmatRichC

Understanding pattern is interesting and looks certainly quicker. Thanks much for the awesome solution.
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For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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Senthil1981 wrote:
For Consecutive integers x, y and z, where x > y > z, which of the following CANNOT be the value of $$(x^2 - y^2)*(y^2 - z^2)$$ ?

A) 63
B) 99
C) 195
D) 276
E) 323

looking at consecutive sequences 3,2,1 and 4,3,2,
(x^2-y^2)*(y^2-z^2)=(x+y)*(y+z)
because both x+y and y+z are sums of odd plus even or even plus odd integers, they must be odd
two odd factors cannot produce an even product
276 cannot be the value (x+y)*(y+z)
D
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Re: For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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Simple question on even/odd concept. D would be the answer
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Re: For Consecutive integers x, y and z, where x > y > z, which of the  [#permalink]

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