Proving Prime p≥3 Satisfies pr ≡ 1, 5, 7 or 11 (mod 12)

[Amannequin]

Homework Statement

If p is a prime and p>3, show that p^r\equiv1,5,7 or 11 (mod12)

Homework Equations

The Attempt at a Solution

Do I go about this by knowing that any prime p greater than 3 is of the form 6n+1 or 6n+5? Any direction on how to go about this will be helpful. Thanks.

 

[Dick]

Homework Statement

If p is a prime and p>3, show that p^r\equiv1,5,7 or 11 (mod12)

Homework Equations

The Attempt at a Solution

Do I go about this by knowing that any prime p greater than 3 is of the form 6n+1 or 6n+5? Any direction on how to go about this will be helpful. Thanks.

Show any other possibility doesn't work. For example, suppose $p^r\equiv 2 (mod 12)$. What's wrong with that?

 

[brmath]

If k is an odd number, list the possibilities for k mod 12.
Next if p is a prime > 3, list the possibilities for p mod 12.
Then what conclusion can you draw about $p^r$?