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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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Updated on: 15 Dec 2012, 00:30
15
00:00

Difficulty:

75% (hard)

Question Stats:

54% (02:07) correct 46% (02:14) wrong based on 485 sessions

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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

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Originally posted by daviesj on 14 Dec 2012, 23:53.
Last edited by daviesj on 15 Dec 2012, 00:30, 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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15 Dec 2012, 05:38
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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.

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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15 Dec 2012, 00:30
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Re: If x and y are positive integers greater than 1 such that x  [#permalink]

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16 Dec 2012, 04:22
Ans:
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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16 Dec 2012, 12:20
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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Re: If x and y are positive integers greater than 1 such that x  [#permalink]

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12 Jun 2013, 05:25
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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

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08 Apr 2016, 15:16
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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Re: If x and y are positive integers greater than 1 such that x  [#permalink]

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09 Apr 2016, 01:50
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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Re: If x and y are positive integers greater than 1 such that x  [#permalink]

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01 Mar 2017, 18:25
Posting official solution of this problem :-
Attachments

official_5.PNG [ 101.34 KiB | Viewed 3850 times ]

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

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29 Sep 2018, 18:10
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

Since x/y is even and y > 1, then x must be an even integer > 2, which can never be a prime integer. Roman numeral I is correct.

Since x - y is even, x + y must be even, too, since x and y are either both odd or both even. Since x and y are both greater than 1, then x + y must be an even integer > 2, which can never be a prime integer. Roman numeral II is correct.

Since x/y is a positive even integer, y/x is the reciprocal of a positive even integer, so y/x is not an integer. Roman numeral III is not correct.

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

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30 Sep 2018, 21:13
Hi All,

We're told that X and Y are positive integers GREATER than 1 such that (X - Y) is an EVEN integer and X/Y is an EVEN integer. We're asked which of the following numbers must be NON-PRIME integers. This question can be solved with a mix of Number Property rules and TESTing VALUES.

To start, we should define the Number Properties that apply to the information in the prompt:
1) Since X and Y are positive integers GREATER than 1 - and (X - Y) is EVEN, X and Y are either BOTH even (re: 4 - 2 = 2) or BOTH odd (7 - 3 = 4).
2) Since X/Y is an EVEN integer, X MUST be an EVEN MULTIPLE of Y. For example 4/2 = 2, 12/3 = 4, 36/6 = 6, etc.

Based on these Number Properties, we can work through the Roman Numerals rather quickly:

I. X

Since Y is GREATER than 1 and X is an EVEN MULTIPLE of Y, we know that X will be 4 or a greater even number. Thus, it will NEVER be a prime.
Roman Numeral 1 is part of the correct answer.

II. (X+Y)

Since X and Y will either be BOTH even or BOTH ODD, the sum will either be (even + even = even) or (odd + odd = even). Thus, this sum will NEVER be a prime.
Roman Numeral 2 is part of the correct answer.

III. Y/X

Since X will ALWAYS be bigger than Y, this fraction will always be a fraction that is less than 1. Therefore, while it won't be a prime number, it won't be an INTEGER either.
Roman Numeral 3 is NOT part of the correct answer.

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

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10 Oct 2019, 00:48
This is a ‘Must be’ kind of a question. We will have to take some simple cases to ascertain which statement can be proved false. The statement/s which cannot be proved false at all will feature in our answers.

The question says that x and y are positive integers greater than 1. This means that neither x nor y can be equal to either 0 or 1. The smallest value that either of these two values can take will be 2.

Also, x-y = even. Since x and y are positive integers, x-y being even only means that x is greater than y. Therefore, x cannot be equal to 2, it HAS to be greater than 2. Also, both x and y HAVE to be even since x/y is given to be even as well.

From all this, we may safely say that the smallest possible value for x HAS to be 4. 4 is a non-prime integer. Since the smallest value of x is non-prime, the other values of x will also be non-prime. Therefore, statement I is always true. Based on this, we can eliminate options B and C since they do not contain statement I in them.

Since both x and y are even and x is at least 4, x+y HAS to be an even number greater than 4 and so cannot be prime. Statement II is always true. Option A can be eliminated.

If y = 2 and x = 4, x-y = 2 and x/y = 2. However, y/x is a fraction and cannot be a non-prime integer. Statement III is not always true. Option E is eliminated.

The correct answer option is D.

Hope this helps!
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Re: If x and y are positive integers greater than 1 such that x   [#permalink] 10 Oct 2019, 00:48
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