# Prime Pairs riddle

Pairs of primes numbers separated by a single number are called prime pairs like 17 and 19. Prove that the number between a prime pair is always divisible by 6 (assuming both numbers in the pair are greater than 6).

Suppose we have 3 consecutive numbers A, B and C where A and C are prime.

Because A and C are primes they are also odd, if they were even they could be divided by two and wouldn’t be prime. This also means that B is even.

Suppose X is the largest multiple of 3 less than A. There could be two possibilities for A: A is X+1 or A is X+2. (If A is X+3, a could be divided by 3 and wouldn’t be prime)

If A is X+1 than B is X+2 and C is X+3. If C is X+3 it can be divided by 3 and wouldn’t be prime so A isn’t X+1.

Which means that A is X+2, B is X+3 and C is X+4. And if B is X+3, B can be divided by 3.

So B can be divided by 2 (even) and by 3, and any number that can be divided by 2 and by 3 can also be divided by 6.

### Your Answer

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