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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]  06 Jan 2013, 07:04
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Question Stats:

45% (02:33) correct 55% (01:08) wrong based on 58 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]  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.
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Kudos [?]: 442 [2] , given: 182

Re: Are the positive integers x and y consecutive? [#permalink]  06 Jan 2013, 22:24
2
KUDOS
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.

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Re: Are the positive integers x and y consecutive? [#permalink]  07 Jan 2013, 02:55
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

Similar question to practice: if-x-y-and-z-are-positive-integers-where-x-y-and-z-x-126859.html
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Re: Are the positive integers x and y consecutive? [#permalink]  30 Sep 2014, 23:21
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Re: Are the positive integers x and y consecutive?   [#permalink] 30 Sep 2014, 23:21
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