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

Question

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

Bunuel · Accepted Answer

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

daviesj · Answer

sorry...edited with right answer...

:)

priyamne · Answer

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).

karanamin · Answer

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.

Bunuel · Answer

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Theory on Number Properties: math-number-theory-88376.html

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kzivrev · Answer

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

Bunuel · Answer

Non-prime integers  are   integers   which are not prime. Can you please share OG question you are talking about? Thank you.

anairamitch1804 · Answer

Posting official solution of this problem :-

ScottTargetTestPrep · Answer

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.

Answer: D

EMPOWERgmatRichC · Answer

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. 
Eliminate Answer B and C.

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. 
Eliminate Answer A.

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. 
Eliminate Answer E.

Final Answer:  D

GMAT assassins aren't born, they're made, 
Rich

CrackverbalGMAT · Answer

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!

Kinshook · Answer

Asked: 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?

x & y are both even integers.

I. x; x/y; x>2 and even ; non-prime
II. x + y; x+y > 2 and even ; non-prime
III. y/x; Since y<x; y/x is a fraction; Not an integer

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

IMO D

AVMachine · Answer

Even No. = 2n (Where n >= 0)

x-y = even(2n) (From here it can be concluded that both x and y are even numbers)

x/y = even(2n); Since y is also even y = 2m; x = 2m*2n = 4mn;

So, x itself is a non-prime no. and x+y = 4mn+2m = 2m(2n+1); which would be also a non-prime

but y/x = 2m/4mn = 1/2n; which won't be an integer.

Hence, answer is D.

sujoykrdatta · Answer

x and y are integers and x - y is even => both even or both odd
x/y is also even => x must be even
Combining: x and y are both even. Thus, x cannot be 2 or only a multiple of 2 since x/y can only be an odd integer 
(example: 6/2 = 3; 2/2 = 1, etc)
Thus, x must be divisible by 4 or higher powers of 2 and y is even.

Thus, both x and x+y are not prime since both are even numbers, greater than 2 
Also, y/x must be a fraction => not an integer

Answer D

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards