Question

For every prime p > 3, 12|(p2 − 1).

Answer 1

Answer:

Proved

Step-by-step explanation:

To prove that for every prime p>3 12 divides $p^2-1$

Proof:

Consider $p^2-1=(p+1)(p-1)$

Since p is prime p cannot be even.  When p is odd, we have p+1 and p-1 as even number.

This gives us the both p+1 and p-1 are divisible by 2, hence product is divisible by 4

To prove the term is divisible by 3:

We have p-1,p, p+1 as consecutive integers hence any one must be divisible by 3.  Since p is prime only either p-1 or p+1 is divisible by3

Hence we have product is divisible by 3 and 4

i.e. 12 divides $p^2-1$, for all prime p >3