Prove that if is an odd prime, prove that either or is divisible by 10.
My proof so far:
If , then we are done. Assume 10 doesn't divide .
Let p=2n+1 for some integer n. Then so 2 can divide it.
Now examine each of these forms.
1) this is never prime.
2) this is only prime for , but the prime is odd so it cannot have this form.
3) since have no common factors this is a possible prime.
4) this is never prime.
5) this is only prime at and that prime is which cannot be because .
6) this is never prime.
7) a possible prime.
8) this is never prime.
9) a possible prime.
Which means has one of three forms: . If you substitute that into you will see it is divisible by . For example, so is divisible. And so on.
I just realized I forgot - but you get the idea. And do not be afraid of this argument, it might look long but that is because I explained it in detail.