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If x and y are positive integers greater than 1 such that x

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If x and y are positive integers greater than 1 such that x [#permalink]

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If x and y are positive integers greater than 1 such that x – y and x/y are both even integers, which of the following numbers must be non-prime integers?

I. x
II. x + y
III. y/x

(A) I only
(B) II only
(C) III only
(D) I and II
(E) I, II and III
[Reveal] Spoiler: OA

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Last edited by daviesj on 14 Dec 2012, 23:30, edited 1 time in total.
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Re: If x and y are positive integers greater than 1 such that x [#permalink]

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New post 14 Dec 2012, 23:10
Am I missing something over here? If x/y is an positive even integer that means x>y then how y/x is an integer?
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Re: If x and y are positive integers greater than 1 such that x [#permalink]

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New post 14 Dec 2012, 23:30
sorry...edited with right answer...:):)
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Re: If x and y are positive integers greater than 1 such that x [#permalink]

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New post 15 Dec 2012, 04:38
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daviesj wrote:
If x and y are positive integers greater than 1 such that x – y and x/y are both even integers, which of the following numbers must be non-prime integers?

I. x
II. x + y
III. y/x

(A) I only
(B) II only
(C) III only
(D) I and II
(E) I, II and III


\(x-y=even\) --> either both even or both odd

\(\frac{x}{y}=even\) --> either both even or \(x\) is even and \(y\) is odd. Also, this statement implies that \(x>y\).

As both statements are true --> both \(x\) and \(y\) must be even.

As \(\frac{x}{y}\) is an even integer and \(y\) is even too, then \(x\) must be multiple of 4 (x/y=even --> x/even=even --> x=even*even).

So, we have that x>y, x is a multiple of 4 and y is even.

Only I and II will give non-prime integers, while III won't be an integer at all.

Answer: D.

Similar question to practice: if-a-and-b-are-positive-integers-such-that-a-b-and-a-b-are-88108.html
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Re: If x and y are positive integers greater than 1 such that x [#permalink]

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New post 16 Dec 2012, 03:22
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from the conditions given to us we get x>y and the min value of y can be 2 , therefore x will be a multiple of 4 so both I and ii will always be non-prime but iii will not be an integer. Answer (D).
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Re: If x and y are positive integers greater than 1 such that x [#permalink]

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New post 16 Dec 2012, 11:20
x - y is an even integer: Therefore, for the equation to hold true, both x and y have to be even.

x/y is an even integer: Therefore, this implies that x has to be greater than y.
We can eliminate option III as y/x will never be an integer (Since x>y)

Statement I, x: x>y, hence x has to greater than 2 and even, therefore divisible by 2 and non-prime.
Statement II, x+y: x+y will be divisible by at least 2 since both x and y are even, hence the sum is non prime.

Thus, option D is the correct answer.
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New post 08 Apr 2016, 14:16
Hi Bunuel or any other expert...

I have a small question.
I answered E which is all 3, I got I and II, but for III of course it will be a fraction and fraction is not a prime number so I includded also III as must be true answer choice. Im I wrong or the question has to add something like a integer and non-prime, something on that line. Im sure you got my point.
Why I think this is beacuse as I remmeber well there was one OG question saying non even integer, and the correct choice was includong odd numebrs and fractions. so I got confused. I perfectly fine arrive to teh answer but this small ambiguity present in the question take away from me the correct answer.

Thanks
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New post 09 Apr 2016, 00:50
kzivrev wrote:
Hi Bunuel or any other expert...

I have a small question.
I answered E which is all 3, I got I and II, but for III of course it will be a fraction and fraction is not a prime number so I includded also III as must be true answer choice. Im I wrong or the question has to add something like a integer and non-prime, something on that line. Im sure you got my point.
Why I think this is beacuse as I remmeber well there was one OG question saying non even integer, and the correct choice was includong odd numebrs and fractions. so I got confused. I perfectly fine arrive to teh answer but this small ambiguity present in the question take away from me the correct answer.

Thanks


Non-prime integers are integers which are not prime. Can you please share OG question you are talking about? Thank you.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: If x and y are positive integers greater than 1 such that x   [#permalink] 09 Apr 2016, 00:50
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