Is the two digit positive integer P a prime number?

1) P + 2 and P – 2 are prime.

2) P – 4 and P + 4 are prime.

Statement (1): P + 2 and P – 2 are prime.

One out of 3 consecutive odd integers, (P – 2), P, and (P + 2) will definitely be a multiple of ‘3’. If (P + 2) and (P – 2) are prime, then P has to be a multiple of ‘3’, which is not prime. The only exception is if the 3 consecutive odd numbers are 3, 5 and 7. However, we are dealing with two digit positive integers.

SUFFICIENT.

Statement (2): P – 4 and P + 4 are prime.

One out of 3 consecutive odd integers, (P – 4), P, and (P + 4) will definitely be a multiple of ‘3’. If (P + 4) and (P – 4) are prime, then P has to be a multiple of ‘3’, which is not prime. The only exception is if the 3 consecutive odd numbers are 3, 7 and 11. However, we are dealing with two digit positive integers.

SUFFICIENT.

Hence, the correct answer is D.

Shreyas Ramakrishnan says

Since it is given that the number is a two digit integer, both statements are independently sufficient. D