OFFICIAL EXPLANATION:

If \(x\) and \(y\) are positive integers, is \(x\) odd?

(1) \(x - 2y\) is a prime number. If \(x - 2y=even \ prime\), so if \(x - 2y=2\), then \(x = even+2y=even+even=even\) (for example \(x=4\) and \(y=1\)), but if \(x - 2y=odd \ prime\), then \(x = odd+2y=odd+even=odd\) (for example \(x=5\) and \(y=1\)). Not sufficient.

(2) \(x + 2y\) is a prime number. Since \(x\) and \(y\) are positive integers, then \(x + 2y\) cannot equal to the only even prime number 2 (because \(x + 2y = \{at \ least \ 1\} + \{ at \ least \ 2\} = \{at \ least \ 3\} > 2\)). Therefore, \(x + 2y=odd \ prime\), which means that \(x =odd \ prime-2y=odd-even=odd\). Sufficient.


Answer: B
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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.

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