Pairs of primes separated by a single number are called prime pairs. Examples are 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).

(From http://www.ocf.berkeley.edu/~wwu/riddles/intro.shtml )

Suppose x+1 and x-1 are both primes greater than 6. If x is odd, then x+1 and x-1 will be even. Therefore, x is even. If x is not divisible by 3, then either x+1 or x-1 will be divisible by 3. Therefore, x is divisible by 3. Since x is divisible by 2 and 3, x is divisible by 6.

Posted by Math Man on 2013-01-24 20:37:24