Number Puzzle #8

The four numbers A, B, A+B and A-B are all prime.  The sum of these four numbers is

A) Even
B) Divisible by 3
C) Divisible by 5
D) Divisible by 7
E) Prime

Source: 2002 AMC 10/12B #15

16 thoughts on “Number Puzzle #8

  1. Yah, yah. Which brings me to a question my students were asking. Pi Day is traditionally celebrated on 3-14 and more specifically at 1:59 (generally PM). BUT, in the year 2015, with the date itself being 3-14-15, will the time be ‘officially’ changed to 9:26? Or will there have to be some international symposium where we all vote on that?

  2. Considering that a large portion of the international community wouldn’t recognize 3-14-15 but instead 14-3-15, it’s going to be hard to get a consensus on the matter. But you make a good point. ;)

  3. WoopS! Forgot to count my knuckles….There aren’t 31 days in April, are there? May 1st would be pretty great. It’s already a federal holiday in most European countries, isn’t it? This could be a good ploy to get Pi Day declared a national holiday.

  4. Got it. Easy as pi/pie. A + B can be prime as long as you remember that there is one prime even number.

  5. Since A, B, A+B and A-B are all primes, we know they are all integers > 1.

    A+B is only prime if either A or B is 2, otherwise odd+odd = even (and > 2) and therefore not prime. They can’t both be 2, as 2+2=4 (not prime).

    A-B is prime, so A>B, so B must be 2, A must be odd (all primes > 2 are odd…), and A >= 5 so that A-B >= 2.

    A + 2 + (A+2) + (A-2) = A + 2 + 2A = 3A + 2.

    Since A is odd, 3A + 2 is odd.

    So the sum is odd, and we can dismiss answer (A). 3A+2 is not divisible by 3, so we can dismiss (B).

    Since B is 2, then the sequence A-2, A, A+2 is a triplet of primes, like 3,5,7. But if A-2 is not divisible by 3, then either A-1 or A is. Assuming it is A-1, then A-1+3=A+2 must be divisible by 3. The result is that one of these three primes A-2,A,A+2 must be 3 itself. Since 1 is not prime, the triplet must be 3,5,7.

    So now we know that A is 5 and B is 2, then the sum is 5+2+3+7 = 17, which is prime.

    Answer: E.

    What is the short version?

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