If x and y are Positive Integers, is x Odd?

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

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“If x and y are positive integers, is x odd?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Answer:

Approach Solution 1

Firstly, take statement (1), we have x – 2y is a prime number
If x – 2y is = even number
So, x – 2y = 2
Then, x = even + 2y = even + even = even
But if x – 2y = odd prime, then x = odd + 2y = odd + even = odd
Hence, this statement is not sufficient in itself.

Now taking statement (2), we have x + 2y is a prime number
As x and y are positive integers, the
x + 2y cannot be equal to only even prime number 2
This is because
x + 2y = {atleast 1} + {atleast 2} = {atleast 3} > 2
Therefore,
x + 2y = odd prime
This means x = odd prime – 2y = odd – even = odd
Hence, this statement is sufficient in itself.

Correct option: B

Approach Solution 2

Taking statement (1), we have:
x – 2y = p
x = p + 2y
p = 2, y = 1 (value can be anything but at the end it would be multiple of 2)
x = 2 + 2 = 4
p = 3, y = 1
x = 3 + 2 = 5
Equation satisfies both the prime number which are even and odd.
Hence, this statement is not sufficient.

Now, taking statement (2), we have:
x + 2y = p
x = p – 2y
p = 2, y = 1
x = 2 – 2 = 0
cannot have prime as 2, as x and y are positive integers.

p = 3, y = 1
x = 3 – 2 = 1
p = 5, y = 1
x = 5 – 2 = 3

As we can see that only even prime number 2 doesn’t satisfies the equation.
But anything after the prime number 2 such as 3, 5, 7 are only odds.
Hence, this statement is sufficient in itself.

Correct option: B

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