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# x and y are positive integers. If x is a prime factor of y^2

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Manager
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x and y are positive integers. If x is a prime factor of y^2 [#permalink]

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21 Apr 2012, 22:05
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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 <=1
E. x+1 is also the prime factor of y^2

GMAT Club Tests: m02 q17
[Reveal] Spoiler: OA

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Posts: 43867
x and y are positive integers. If x is a prime factor of y^2 [#permalink]

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21 Apr 2012, 22:10
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If $$x$$ and $$y$$ are positive integers and $$x$$ is a prime factor of $$y^2$$, then which of the following MUST be true?

A. $$x$$ is even
B. $$x=y^2$$
C. $$\frac{y}{x}$$ is even
D. $$\frac{x}{y}\leq{1}$$
C. $$x+1$$ is also a prime factor of $$y^2$$

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. $$\frac{y}{x}$$ is even --> not necessarily true: if $$x=3$$ and $$y=3$$ then $$\frac{y}{x}=1=odd$$;
D. $$\frac{x}{y}\leq{1}$$ 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 $$y^2=4$$.

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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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14 Jan 2017, 19:10
x,y Positive Integers
x = prime factor of y^2
Try numbers
1, Y = 3 ; x is 3
2, Y = 4 ; X = 2
3, Y =12 ; X = 2,3
Eliminate A,B,C
For any value x/y <= 1
D
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Posts: 16
Re: x and y are positive integers. If x is a prime factor of y^2 [#permalink]

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14 Jan 2017, 22:25
If x is prime factor of y^2, then x is a factor of y. Therefore x<=y which leads to x/y<=1. One can safely divide both sides by y because x and y are positive.

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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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18 Jan 2018, 13:18
monir6000 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 <=1
E. x+1 is also the prime factor of y^2

Let’s go through each answer choice:

A. x is even

If we let x = 3 and y = 3, we can see that x is a prime factor of y^2. So x does not need to even. Choice A is not correct.

B. x = y^2

Using the same numbers for x and y as in choice A, we see that choice B is not correct also.

C. y/x is even

Using the same numbers for x and y as in choice A, we see that choice C is not correct also.

D. x/y <=1

Using the same numbers for x and y as in choice A, we see that choice D might be correct. Let’s examine choice E.

E. x+1 is also the prime factor of y^2

Using the same numbers for x and y as in choice A, we see that choice D can’t be correct since x + 1 = 4 is not a prime factor y^2 = 9.

Thus, the correct answer must be D.

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Re: x and y are positive integers. If x is a prime factor of y^2   [#permalink] 18 Jan 2018, 13:18
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