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

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X and Y are positive integers. If X is a prime factor of Y^2 [#permalink]

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05 Jan 2011, 03:57
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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]

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05 Jan 2011, 05:55
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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.

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 $$y^2=4$$.

Hope it's clear.
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Re: M02 Q17 [#permalink]

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05 Jan 2011, 06:46
Thanks Bunuel.
You are genius.
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Re: M02 Q17 [#permalink]

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06 Jan 2011, 03:26
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]

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10 Oct 2014, 08:11
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Re: X and Y are positive integers. If X is a prime factor of Y^2 [#permalink]

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25 May 2016, 00:59
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Re: X and Y are positive integers. If X is a prime factor of Y^2   [#permalink] 25 May 2016, 00:59
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# X and Y are positive integers. If X is a prime factor of Y^2

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