If x, y, and z are positive integers, where x > y and

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If x, y, and z are positive integers, where x > y and [#permalink]

15 Nov 2009, 22:47

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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

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Re: Are x and y consecutive perfect squares [#permalink]

16 Nov 2009, 01:27

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ctrlaltdel wrote:

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

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Re: Are x and y consecutive perfect squares [#permalink]

16 Nov 2009, 01:42

[Reveal] Spoiler: OA

B
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Re: Are x and y consecutive perfect squares [#permalink]

16 Nov 2009, 07:55

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

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Re: Are x and y consecutive perfect squares [#permalink]

16 Nov 2009, 08:40

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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.

Answer: B.
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Director

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Re: Are x and y consecutive perfect squares [#permalink]

16 Nov 2009, 10:08

Thanks Bunuel

! Great job..+1K

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Re: Are x and y consecutive perfect squares [#permalink]

17 Nov 2009, 20:07

bunuel - sometimes i wonder how easily you solve these questions - you are awesome

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Re: Are x and y consecutive perfect squares [#permalink] 17 Nov 2009, 20:07

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If x, y, and z are positive integers, where x > y and

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If x, y, and z are positive integers, where x > y and

If x, y, and z are positive integers, where x > y and

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