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 Prime Pairs (Posted on 2002-09-01)
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).

 See The Solution Submitted by levik Rating: 3.3636 (11 votes)

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 Solution | Comment 7 of 8 |

Let the prime pairs be denoted by (a, b), with a< b, and a> 6, are:
Being primes, both a and b must be odd.

Then, the possible values for (a,b) are:

(i) (a, b) (mod 6) = (1,3)

(ii) (a, b) (mod 6) = (3,5)

(iii) (a, b) (mod 6)= (5,7) (mod 6) = (-1,1)

But in (i), b (mod 6) = 3, is always divisible by 3, for b>6 and in (ii), a (mod 6) = 3, is always divisible by 3, for a>6. This is a contradiction.

Thus,  (a, b) (mod 6) = (-1,1), so that: (a+1) mod 6 = (b-1) mod 6 = 0

Hence, the proof.

Note: For a < 6, the pair a=5, b=7 also satisfy the given conditions.

 Posted by K Sengupta on 2008-11-06 05:47:18

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