What do you notice about remainders when prime number are divided by 6?

Answer 1

If the primes are 5 or greater, then the remainders are 1 or 5.

This is so trivially obvious.
The remainder cannot be 0 or else the number is divisible by 6 and so not a prime.

The remainder cannot be 2 or 4 or else the number is divisible by 2 and so not a prime.

The remainder cannot be 3 or else the number is divisible by 3 and so not a prime.

That just leaves 1 and 5: 11 leaves a remainder of 5, 13 leaves 1 for example.

Answer 2

They do not take the values 0 or 4, but can take all other values.