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Manager  Joined: 24 Jul 2009
Posts: 65
Location: United States
GMAT 1: 590 Q48 V24 If x, y, and z are positive integers, where x > y and  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 39% (02:50) correct 61% (03:00) wrong based on 353 sessions

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If x, y, and z are positive integers, where x > y and z=x^(1/2), are x and y consecutive perfect squares? (A perfect square is defined as the square of an integer. For example, 36 is a perfect square since it equals 6 squared, while 38 is not a perfect square since it is not equal to the square of any integer.)

(1) x + y = 8z +1
(2) x – y = 2z – 1

Originally posted by ctrlaltdel on 15 Nov 2009, 22:47.
Last edited by Bunuel on 13 Jun 2013, 05:51, edited 1 time in total.
Math Expert V
Joined: 02 Sep 2009
Posts: 58427
Re: Are x and y consecutive perfect squares  [#permalink]

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17
7
Economist wrote:
What is the source?? This is a heck of a problem. Plugging numbers is painful. Do we have any algebraic approach?

Here you go.
If $$x$$ and $$y$$ are consecutive perfect squares, then $$\sqrt{y}$$ and $$\sqrt{x}$$ must be consecutive integers.

So, the question: is $$\sqrt{y}=\sqrt{x}-1$$?

Square both sides: $$y=x-2\sqrt{x}+1$$?

(1) $$x+y=8z+1$$
$$x+y=8\sqrt{x}+1$$
$$y=8\sqrt{x}+1-x$$

So basically question transforms to is $$8\sqrt{x}+1-x=x-2\sqrt{x}+1$$?
$$10\sqrt{x}=2x$$?
$$5\sqrt{x}=x$$?
$$\sqrt{x}=5$$?
$$x=25$$?

If x=25 then yes, if not then no. But we don't know the value of x, hence insufficient.

(2) $$x-y=2\sqrt{x}-1$$
$$y=x-2\sqrt{x}+1$$, which is exactly what we are asked in the question. Hence sufficient.

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Manager  Joined: 30 Aug 2009
Posts: 227
Location: India
Concentration: General Management
Re: Are x and y consecutive perfect squares  [#permalink]

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2
1
ctrlaltdel wrote:
If x, y, and z are positive integers, where x > y and z=x^(1/2), are x and y consecutive perfect squares? (A perfect square is defined as the square of an integer. For example, 36 is a perfect square since it equals 6 squared, while 38 is not a perfect square since it is not equal to the square of any integer.)

(1) x + y = 8z +1
(2) x – y = 2z – 1

Happy Solving

B

1. x+ y = 8z+1
the values of x and y satisfying this equation is 25 and 16 which are consecutive perfect squares
the values of x and y satisfying this equation is 64 and 1 which are NOT consecutive perfect squares
Hence insuff

2. x -y = 2z-1
the values x and y satisfying this equation will be all consecutive perfect squares (be it [4,1] [9,4][2500,2401]
non consecutive perfect squares will not satisfy the equation
hence suff
Manager  Joined: 24 Jul 2009
Posts: 65
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GMAT 1: 590 Q48 V24 Re: Are x and y consecutive perfect squares  [#permalink]

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Spoiler: :: OA
B
Kudos to kp1811
Director  Joined: 01 Apr 2008
Posts: 650
Name: Ronak Amin
Schools: IIM Lucknow (IPMX) - Class of 2014
Re: Are x and y consecutive perfect squares  [#permalink]

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What is the source?? This is a heck of a problem. Plugging numbers is painful. Do we have any algebraic approach?
Director  Joined: 01 Apr 2008
Posts: 650
Name: Ronak Amin
Schools: IIM Lucknow (IPMX) - Class of 2014
Re: Are x and y consecutive perfect squares  [#permalink]

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Thanks Bunuel ! Great job..+1K
Manager  Joined: 16 Apr 2009
Posts: 214
Re: Are x and y consecutive perfect squares  [#permalink]

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bunuel - sometimes i wonder how easily you solve these questions - you are awesome _________________
Always tag your question
SVP  Joined: 06 Sep 2013
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Concentration: Finance
Re: If x, y, and z are positive integers, where x > y and  [#permalink]

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IM BACK!

The difference of perfect squares can be expressed as an odd number. Or viceversa, an odd number can be expressed as the difference of two consecutive perfect squares

Namely,

(k+1)^2 - (k)^2 = 2k + 1

Hope this clarifies
Manager  Joined: 03 May 2013
Posts: 67
Re: If x, y, and z are positive integers, where x > y and  [#permalink]

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If x, y, and z are positive integers, where x > y and z=x^(1/2), are x and y consecutive perfect squares? (A perfect square is defined as the square of an integer. For example, 36 is a perfect square since it equals 6 squared, while 38 is not a perfect square since it is not equal to the square of any integer.)

(1) x + y = 8z +1
(2) x – y = 2z – 1

Alt way

2. z^2-2z+1 =y
(z-1)^2 = y, z-1 = y^1/2, z=y^1/2+1, since z=X^1/2 , x^1/2 and y^1/2 are consecutive
1. z^2-8z-1 =-y NO solution
Hence B
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Joined: 20 Jun 2013
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GMAT 1: 430 Q39 V25 GPA: 3.5
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Re: If x, y, and z are positive integers, where x > y and  [#permalink]

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thanks that was a good one... pretty weak in ds got a get the basics first i guess
Non-Human User Joined: 09 Sep 2013
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Re: If x, y, and z are positive integers, where x > y and  [#permalink]

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_________________ Re: If x, y, and z are positive integers, where x > y and   [#permalink] 30 Sep 2018, 10:15
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