### Instruction: The positive integers $A,$ $B,$ $A-B,$ and $A+B$ are all prime numbers. The sum of these four primes is

$\bullet$ A. even

$\bullet$ B. divisible by $3$

$\bullet$ C. divisible by $5$

$\bullet$ D. divisible by $7$

$\bullet$ E. prime

Express your answer using a letter, as A, B, C, D, or E.### Response: I need to find four primes of the form $A,$ $B,$ $A-B,$ and $A+B.$
I notice that $A-B$ and $A+B$ are both even, except when $B=2.$
So, $B$ must be $2,$ otherwise I would have two even primes, which is impossible.
Now, I have $A,$ $2,$ $A-2,$ and $A+2$ are all primes.
The smallest such $A$ is $5,$ which gives me $5, 2, 3, 7$ as the four primes.
The sum of these four primes is $17,$ which is also prime.
Therefore, the answer is E. prime.