2002 AMC 10B Problem 15

The numbers $A - B$ and $A + B$ differ by $2B,$ so they have the same parity. Being prime, they must both be odd, which forces $A$ and $B$ to have opposite parity. Since $2$ is the only even prime, either $A=2$ or $B=2.$ The first case is impossible because the positive prime $B$ would make $A-B<0.$ Hence $B=2.$ Now $A - 2,$ $A,$ and $A + 2$ are three primes. One of any three integers spaced $2$ apart is divisible by $3,$ so that member must itself be $3.$ The only positive possibility is $3,$ $5,$ $7.$ The four primes are $2,$ $3,$ $5,$ $7,$ and their sum is $17,$ which is prime. Thus, the correct answer is E.