If x and y are positive integers, is x even?

Question

If x and y are positive integers, is x odd?

(1) x - 2y is a prime number
(2) x + 2y is a prime number

M33-103

Bunuel · Accepted Answer

If  x  and  y  are positive integers, is  x  odd? 
 
(1)  x - 2y  is a prime number. 
 
If  x - 2y=even \ prime , that is if  x - 2y=2 , then  x = even+2y=even+even=even  (consider  x=4  and  y=1  as an example). However, if  x - 2y=odd \ prime , then  x = odd+2y=odd+even=odd  (consider  x=5  and  y=1  as an example). Not sufficient.
 
(2)  x + 2y  is a prime number. 
 
Given that  x  and  y  are positive integers, it's impossible for  x + 2y  to equal 2, the only even prime number. This is because  x + 2y = \{minimum \ 1\} + \{ minimum \ 2\} = \{minimum \ 3\} > 2 . Therefore,  x + 2y  must equal an  odd \ prime . In this case,  x =odd \ prime-2y=odd-even=odd . Sufficient.

Answer: B

DinkarDK · Answer

St1
Lets put in values: 
X=7 and y=2 then X-2Y = 3 = prime, X is odd
X = 4 and Y=2 then X-2Y = 2 = prime, X is even
So Statement 1 is not sufficient.

St 2
Now as 2 is the only even prime number and we already have 2Y which is multiple of 2, so any even valve of X will result in event number which will not be a prime number.
Hence X for 2nd case has to be odd for X+2Y to be prime number

Statement 2 is sufficient

Answer: option B

SSM700plus · Answer

Solution :-

x and y are greater than zero.

Statement 1:- x - 2y is a prime number

Let's take some example.

i)  X IS ODD

x= 9 and y=2. x-2y = 9-2(2) = 5 is a prime number. And x is odd is TRUE.

ii)  X IS even 
x= 4 and y = 1. x-2y = 4 - 2(1) = 2 is a prime number. And x is not odd. So FALSE.

It implies even if x-2y is prime number, x can be odd or even.

Hence insufficient.

Statement 2: - x + 2y is a prime number.

We can do this using numbers too. But let's try thinking in terms of odd/even and prime number.

We know that 2 is the only even prime number.

Also it is given that x and y is greater that zero. So let's assume minimum value of y =1.

That means Statement 2 is  x + 2 is prime numbers .

It implies that x cannot be 2. If x =2, then x+2 = 2+2 =4 is not a prime number.

In other words. x has to be odd in order to make x+2y as prime.

Hence Statement 2 is Sufficient.

Ans -B.

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

GMATin · Answer

The point to remember in this question is that x and y are given as positive integers, therefore x and y can take values such as 1, 2, 3...

Now, we need to check if x can be expressed as 2n+1

1) Given x-2y is prime

Here, 2y will be at least 2 (since lowest value y can take is 1)
This shows that x can take values such as 4 or 5:
If x is 4, then x-2y=4-2=2 (which is a prime number).  Thus, x is even 
and 
If x is 5, then x-2y=5-2=3 (which is a prime number).  Thus, x is odd

Since, we cannot conclusive check if x will be even or odd, this is insufficient.

2) Given x+2y is prime

Here, again 2y will be at least 2 (since lowest value y can take is 1)

Now, we need to check  only for values 2 and 3 because all prime numbers that are greater than 2 are odd.

If x+2y=2, x can only take value 0 which is not possible since x has to a positive integer. In fact, x can never take 2 as a value in order to satisfy this equation.

If x+2y=3, then x=1 (which is odd). Thus, we have safely concluded that no even value of x can make x+2y result in a prime number.  Hence, this is sufficient

Answer is B

ramlala · Answer

If x and y are positive integers, is x odd?

(1) x - 2y is a prime number

x may or may not be odd
NOT SUFFICIENT

(2) x + 2y is a prime number

as x and y are +ve, min value is 1 resulted in  > 3. 
As all the prime numbers  > 3 are odd, and 2y is always even, x must be ODD.
SUFFICIENT

IMO B

bumpbot · Answer

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If x and y are positive integers, is x even?

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