Answer:
Proved
Step-by-step explanation:
To prove that for every prime p>3 12 divides [tex]p^2-1[/tex]
Proof:
Consider [tex]p^2-1=(p+1)(p-1)[/tex]
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 [tex]p^2-1[/tex], for all prime p >3
