[INST] The positive integers A, B, A-B, and A+B are all prime numbers. The sum of these four primes is

A. even
B. divisible by 3
C. divisible by 5
D. divisible by 7
E. prime

Express your answer using a letter, as A, B, C, D, or E./n [/INST]  ###Response: We are given that the positive integers A, B, A-B, and A+B are all prime numbers. We need to find the sum of these four primes.

First, note that A-B and A+B are either both odd or both even. However, they are also both prime, so they must both be odd.

Therefore, one of A and B is odd and the other even. Since A is the prime number lying between A-B and A+B, A must be the odd prime.

Thus, B=2 (the only even prime), and A-2, A, and A+2 are consecutive odd primes (3, 5, and 7).

The sum of the four primes 2, 3, 5, and 7 is the prime number 17.

Therefore, the correct answer is (E), prime.