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. Sean says:

Got it. Piece of Cake.

2. On a math blog, you say “Easy as pie”

3. Sean says:

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?

4. 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. ðŸ˜‰

5. Sean says:

6. Hahhaaha! I think it would confuse a lot of people to claim that May 1st was now Pi Day.

7. Sean says:

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.

8. Bo Li says:

The answer is A, for prime are all odd, and odd adds odd equals even!

9. Bo Li says:

The answer for puzzle 8 is A), for all prime are odd, and odd adds odd equals even.

10. Fairly certain the answer is E.

11. Djordje says:

I would say none…

If A and B are prime, A+B can’t be prime.

12. Djordje says:

Woops, I forgot 2… It’s prime then…

13. Hi, nice posts there ðŸ™‚ thank’s for the interesting information

14. Karin says:

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

15. Karin says:

Forgot to say, the answer is E.

16. 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.