Question: If p and q are two different odd prime numbers, such that p < q, then the following must be true?

1. (2q + p) is a prime number
2. p + q is divisible by 4
3. q - p is divisible by 4
4. (p + q + 1) is the difference between two perfect squares of integers
5. \(p^2+q^2\)is the difference between two perfect squares of integers

Answer:

Approach Solution (1):

Fact: Any odd number can be represented as \(2n \pm 1\) , where n is of-course an integer

Scenario I: 2n + 1 =\((n+1)^2-n^2 \rightarrow\) An odd number can be represented as difference of 2 perfect square

Scenario II: 2n – 1 =\(n^2-(n-1)^2 \rightarrow\)An odd number can be represented as difference of 2 perfect square

We know that p, q are both odd integers, thus p + q + 1 = odd + odd + odd = 3 * odd = odd. thus, p +nq + 1 can always be represented as a difference of 2 perfect squares

1. p = 7, q = 11, 2p + q = 25, not prime
2. p = 7, q = 11, p + q is not divisible by 4
3. p = 3, q = 5, p – q is not divisible by 4.
4. Any odd number is always a difference of 2 perfect square. Correct Answer
5. For p = 3, q = 5, we have \(p^2+q^2=34\), note that for the given problem, \(p^2+q^2\)will always be an even integer.

Now, any even integer can be represented as 2n, where n is an integer.
\((n+1)^2-(n-1)^2= 4n = 2*2n\)

Thus, any even number, which is a multiply of 4, can be represented as the difference of 2 perfect squares. As 34 is not divisible by 4, we can safely eliminate E.

Correct Option: D

“If p and q are two different odd prime numbers, such that p < q, then the following must be true?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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