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# If x and y are positive integers and x = y^(a -2), is x a factor of y?

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Manager
Joined: 03 Jul 2017
Posts: 62

Kudos [?]: 47 [0], given: 30

Location: India
Concentration: Finance, Economics
GPA: 3.4
If x and y are positive integers and x = y^(a -2), is x a factor of y? [#permalink]

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24 Nov 2017, 01:31
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95% (hard)

Question Stats:

31% (01:42) correct 69% (01:14) wrong based on 75 sessions

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If x and y are positive integers and $$x=y^{a-2}$$, is x a factor of y ?

(1) a is an integer
(2) a < 4
[Reveal] Spoiler: OA

Last edited by Bunuel on 24 Nov 2017, 01:35, edited 1 time in total.
Renamed the topic and edited the question.

Kudos [?]: 47 [0], given: 30

Math Expert
Joined: 02 Sep 2009
Posts: 42559

Kudos [?]: 135309 [1], given: 12686

Re: If x and y are positive integers and x = y^(a -2), is x a factor of y? [#permalink]

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24 Nov 2017, 01:45
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If x and y are positive integers and $$x=y^{a-2}$$, is x a factor of y ?

Is $$\frac{y}{x}=integer$$?

Is $$\frac{y}{y^{a-2}}=integer$$?

Is $$y^{3-a}=integer$$?

For $$y^{3-a}=integer$$ to be true for any value of y, the exponent must be a non-negative integer. So, the question basically asks whether a is an integer and $$3-a \geq 0$$? Or which is the same: is a an integer and $$a \leq 3$$?

(1) a is an integer. Not sufficient.

(2) a < 4. Not sufficient.

(1)+(2) Sufficient.

_________________

Kudos [?]: 135309 [1], given: 12686

Manager
Joined: 03 Jul 2017
Posts: 62

Kudos [?]: 47 [0], given: 30

Location: India
Concentration: Finance, Economics
GPA: 3.4
Re: If x and y are positive integers and x = y^(a -2), is x a factor of y? [#permalink]

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24 Nov 2017, 02:19
Thank Bunuel for the awesome explanation!

Kudos [?]: 47 [0], given: 30

Intern
Joined: 24 Nov 2016
Posts: 42

Kudos [?]: 3 [1], given: 21

Re: If x and y are positive integers and x = y^(a -2), is x a factor of y? [#permalink]

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24 Nov 2017, 08:03
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Bunuel wrote:
If x and y are positive integers and $$x=y^{a-2}$$, is x a factor of y ?

Is $$\frac{y}{x}=integer$$?

Is $$\frac{y}{y^{a-2}}=integer$$?

Is $$y^{3-a}=integer$$?

For $$y^{3-a}=integer$$ to be true for any value of y, the exponent must be a non-negative integer. So, the question basically asks whether a is an integer and $$3-a \geq 0$$? Or which is the same: is a an integer and $$a \leq 3$$?

(1) a is an integer. Not sufficient.

(2) a < 4. Not sufficient.

(1)+(2) Sufficient.

If anyone wants to know the step by step on how to get $$y^{3-a}$$:

using the exponent rule: $$\frac{x^{a}}{x^{b}}$$;
so, $$\frac{y}{y^{a-2}}$$ is the same as $$\frac{y^1}{y^{a-2}}$$ which becomes, $$y^{1-(a-2)}$$ --> $$y^{1-a+2}$$ finally, $$y^{-a+3}$$.

Kudos [?]: 3 [1], given: 21

Re: If x and y are positive integers and x = y^(a -2), is x a factor of y?   [#permalink] 24 Nov 2017, 08:03
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