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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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New post 21 Apr 2012, 23:05
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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

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

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New post 21 Apr 2012, 23:10
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 \(y^2=4\).

Answer: 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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New post 14 Jan 2017, 20:10
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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Re: x and y are positive integers. If x is a prime factor of y^2  [#permalink]

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New post 14 Jan 2017, 23:25
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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New post 18 Jan 2018, 14:18
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.

Answer: 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, 14:18
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