Prove that if x is a prime number greater than 3, then x%6 = 1 or x%6 = 5

Question

I'm struggling with this Discrete Math problem:

Prove that if x is a prime number greater than 3, then x%6 = 1 or x%6 = 5.

I've tried rewriting x in terms of odd numbers (2k+1), but didn't get far, and the contrapositive doesn't seem much more helpful.

Answer 1

Any number, $x$, can be written as one of the following:

$6k$
$6k+1$
$6k+2$
$6k+3$
$6k+4$
$6k+5$

If $x$ is a prime number, it cannot be $6k$ or $6k+2$ or $6k+3$ or $6k+4$ because they have factors, 6, 2, 3 and 2 respectively.

Therefore, if $x$ is a prime number, it must be one of $6k+1$ or $6k+5$.

Now, $(6k+1)\% 6 = 1$ and $(6k+5)\%6 = 5$.

Hence, the claim is true.

Answer 2

If x is greater than 3, then it cannot be divisible by any number. Therefore, it must be coprime with 6, and the only options are 1 and 5 because 2, 3, and 4 have a gcf > 1 with respect to 6.