Answer
See below.
Work Step by Step
Based on the quotient-remainder theorem, any integer $n$ must be in one of the following forms: n mod 6=0,1,2,3,4,5, or $n=6q, 6q+1,6q+2,6q+3,6q+4,6q+5$.
Instead of proving $6q+1$ or $6q+5$ are prime numbers, we only need to prove that others must not be prime numbers.
Case1: $n=6q$ has factors 2 and 3, thus it can not be a prime number;
Case2: $n=6q+2$ has a factors 2, thus it can not be a prime number;
Case3: $n=6q+3$ has a factors 3, thus it can not be a prime number;
Case4: $n=6q+4$ has a factors 2, thus it can not be a prime number;
Based on the above results, all prime numbers (except 2 and 3) must be in one of the forms of $6q+1$ or $6q+5$.