Why do prime numbers have a remainder of 1 or 5 when divided by 6?

If the number has a remainder of 2 or 4 when divided by 6, it is an even number, so it is not Prime number.

6n + 2 = 2 (3n + 1), so it is divisible by 2.

6n + 4 = 2 (3n + 2), so it is divisible by 2.

If the number has a remainder of 3 when divided by 6, it is divisible by 3, so it is not a prime number.

6n + 3 = 3 (2n + 1), so it is divisible by 3.

However, there are two prime numbers that do not have a remainder of 1 or 5 when divided by 6 : 2 and 3