The positive integers A, B, A - B, and A + B are all prime numbers. Th

Question

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

Bunuel · Accepted Answer

For both A - B and A + B to be primes, B must be 2. This is because both A and B cannot be odd, and both cannot be even (so one must be odd and the other must be the only even prime, which is 2). A cannot be 2 because 2 minus any prime cannot result in another prime.

Thus, we have that B = 2, and A - 2, A, and A + 2 must all be primes.

Now, A - 2, A, and A + 2 are three consecutive odd integers. The only case where three consecutive odd integers are all prime is with 3, 5, and 7 (since one of the three consecutive odd integers must be a multiple of 3, and if they are all primes, that multiple must be 3 itself, resulting in 3, 5, and 7).

A - 2 = 3 implies that A = 5.

The sum of A, B, A - B, and A + B is 3A + B = 17, which is a prime number.

Answer: E.

ankitgoswami · Answer

Take 
A=5
B=2
A+B=7
A-B=3

Adding all we get 17.
A Prime.

Ans is  E .

Posted from my mobile device

Hoehenheim · Answer

Bunuel  Got this using trial and error. But, how to do this algebraically? 
Got down to: sum = 6(3N + n) + (number). where A= 6N +/- 1, B = 6n +/- 1.

When trying to sort out A-B and A+B, I hit a wall, as to make both of those expressions prime, I could not decide on the value of +1 or -1 to be assigned to the variables A and B.

Hoehenheim · Answer

Thanks, I now realize that being unable to pin down a correct expression of both A-B and A+B as primes when both A and B are prime, points to a fault in the 6n +/- 1 way of looking at prime numbers which concurrently means that either A and B have to be 2 or 3 (prime numbers outside the definition of 6n +/- 1) which narrows down the scope of this question. Conversely, we could say that no 4 prime numbers not including 2 would yield such results, at least up to the mark where 6n +/-1 is valid.

messi16 · Answer

For this one, begin by listing out all primes. We know that the most common primes are 1, 2, 3, 5, 7, 11, 13, and 17. From here, identify two primes that satisfy A + B = prime and A - B = prime. After a few test trials, we can see that A= 5 and B = 2 because 5 + 2 = 7 and 5 - 2 = 3, which satisfy our equations. Thus the set is 2, 5, 3, 7, which add up to 17. This is a prime number and thus we can deduce E as our correct answer.

The positive integers A, B, A - B, and A + B are all prime numbers. Th

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The positive integers A, B, A - B, and A + B are all prime numbers. Th

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