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X and Y are positive integers. If X is a prime factor of Y^2

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Status: Preparing for GMAT - March 2011
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X and Y are positive integers. If X is a prime factor of Y^2 [#permalink] New post 05 Jan 2011, 02:57
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X and Y are positive integers. If X is a prime factor of Y^2 , which of the following MUST be true?

A. X is even
B. X=Y^2
C. Y/X is even
D. X/Y is less than or equal to 1
E. X+1 is also a prime factor of Y^2

M02 Q17
[Reveal] Spoiler: OA

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Re: M02 Q17 [#permalink] New post 05 Jan 2011, 04:55
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tirupatibalaji wrote:
X and Y are positive integers. If X is a prime factor of Y^2 , which of the following MUST be true?
(a) X is even
(b) X=Y^2
(c)Y/X is even
(d) X/Y is less than or equal to 1
(c)X+1 is also a prime factor of Y^2

Please need a good explanation for this.
Bunuel ...please help :)


Given: x=prime, y=positive \ integer and x is a prime factor of y^2. Now, as x is a prime factor of y^2 and y is an integer then x must be a factor of y as well (how else could x appear in the prime factorization of y^2=y*y if it's not a factor of y). Next, as x is a factor of y (and y is a positive integer) then x\leq{y} --> \frac{x}{y}\leq{1}, so D must be true.

Another approach:
Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

A. x is even --> not necessarily true: x=3=odd and y=3;
B. x=y^2 --> never true, as x=prime then it can not equal to square of another integer;
C. y/x is even --> not necessarily true: if x=3 and y=3 then \frac{y}{x}=1=odd;
D. x/y is less than or equal to 1 --> always true as shown above or by POE;
E. x+1 is also a prime factor of y^2 --> not necessarily true: if x=2 and y=2 then x+1=3 and 3 is not a factor of y^2=4.

Answer: D.

Hope it's clear.
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Manager
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Status: Preparing for GMAT - March 2011
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Schools: INSEAD, RSM, HEC, St. Gallen, IF, IESE
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Re: M02 Q17 [#permalink] New post 05 Jan 2011, 05:46
Thanks Bunuel.
You are genius.
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Re: M02 Q17 [#permalink] New post 06 Jan 2011, 02:26
Bunuel has explained it well. Ans must be D
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Re: X and Y are positive integers. If X is a prime factor of Y^2 [#permalink] New post 10 Oct 2014, 07:11
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Re: X and Y are positive integers. If X is a prime factor of Y^2   [#permalink] 10 Oct 2014, 07:11
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