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Are the positive integers x and y consecutive?

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Are the positive integers x and y consecutive? [#permalink] New post 06 Jan 2013, 07:04
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43% (02:14) correct 57% (01:19) wrong based on 35 sessions
Are the positive integers x and y consecutive?

(1) x^2 - y^2 = 2y + 1
(2) x^2 - xy - x = 0
[Reveal] Spoiler: OA

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Re: Are the positive integers x and y consecutive? [#permalink] New post 06 Jan 2013, 10:00
Expert's post
daviesj wrote:
Are the positive integers x and y consecutive?
(1)x^2 - y^2 = 2y + 1
(2) x^2 - xy - x = 0


The question is basically asking whether x=y+1
Statement 1)
x^2 - y^2=2y+1 can be written as (x+y)(x-y)=2y+1.----equation 1
If we put x=y+1, then LHS must be equal to RHS.

Equation 1 can be written, after substituting x=y+1, as (2y+1)(1)=2y+1. They are equal. Hence x and y are consecutive.

Statement 2)
x(x-y-1)=0
The above equation can be equal to 0 only when x-y=1 because x is given to be positive.

Hope I am correct.
+1D
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Re: Are the positive integers x and y consecutive? [#permalink] New post 06 Jan 2013, 22:24
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You can simplify the first statement in following way:
(1) x^2-Y^2=2y+1
i.e. x^2 = Y^2 + 2y+1
i.e. x^2 = (y+1)^2
i.e. x=|y+1| = y + 1(taking positive root as both x & y are positive integers)
Hence (1) is SUFFICIENT as x & y are consecutive

(2) x^2-xy-x=0
i.e. x(x-y-1)=0
i.e. x=0 or x=y+1
As x is positive integer x<>0, thus x=y+1
Hence (2) is SUFFICIENT.

Choice (D) is the answer.
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Re: Are the positive integers x and y consecutive? [#permalink] New post 07 Jan 2013, 02:55
Expert's post
Re: Are the positive integers x and y consecutive?   [#permalink] 07 Jan 2013, 02:55
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