Suppose we have p is a prime, a > 1, and gcd(a,p) = 1
1.) Prove that a^(ord_(p)a - 1) + a(ord_(p)a - 2) + ... + a^1 + 1 = 0 (mod p)
2.) Prove that if a has order 3 (mod p), then a + 1 has ord 6 (mod p).
Let us first show that if gcd(a,p)=1 then gcd(a+1,p)=1. Meaning, we want to show that the order modulo p is defined (because if gcd(a+1,p)=p then the order is not defined!)
Say, it is not, gcd(a+1,p)=1 that is a+1=pn. Hence, a=pn-1.
But then a = pn -1 = -1 (mod p) meaning the order is 2 not 3, a contradiction.
Thus, this problem does not lead to an undefined operation.
Now we can solve this problem, gaze below.