SOLUTION: For a prime p>3, prove that p^2

SOLUTION: For a prime p>3, prove that p^2 - 1 is divisible by 12

Question 966690: For a prime p>3, prove that p^2 - 1 is divisible by 12

Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!
For a prime p>3, prove that p^2 - 1 is divisible by 12

For any prime p>3, p^2-1 > 6.

All odd numbers > 6 are between 2 consecutive multiples of 6.

Between every 2 consecutive multiples of 6, there are 3 
consecutive odd numbers, 6k+1, 6k+3, and 6k+5. I.e.,

6k < 6k+1 < 6k+3 < 6k+5 < 6k+6=6(k+1)

Thus every odd number > 6 can be written as 6k+1, 6k+3 or 6k+5.

As all primes greater than 3 are odd, they can all be written 
as 6k+1 or 6k+5, (but never as 6k+3 since that is never prime).

(6k+1)�-1 = 36k�+12k+1-1 = 36k�+12k    <-- a multiple of 12
(6k+5)�-1 = 36k�+60k+25-1 = 36k�+60k+24 <-- a multiple of 12

That proves it.

Edwin

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