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

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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.

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

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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.

Answer: C.
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Kudos [?]: 135309 [1], given: 12686

Manager
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Joined: 03 Jul 2017
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Kudos [?]: 47 [0], given: 30

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Concentration: Finance, Economics
GPA: 3.4
GMAT ToolKit User Premium Member CAT Tests
Re: If x and y are positive integers and x = y^(a -2), is x a factor of y? [#permalink]

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

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

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New post 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.

Answer: C.


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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