>6n is a multiple of 6
>6n + 1 could be prime
>6n + 2 = 2(3n + 1) is even
>6n + 3 = 3(2n + 1) is a multiple of 3
>6n + 4 = 2(3n + 2) is even
>6n + 5 could be prime
>6n + 6 = 6(n + 1) is the next multiple of 6.
Therefore each prime (except 2 and 3) is next to a multiple of 6 and that too if that number leaves 1 or 5 as remainder on being divided by 6.
>6n + 1 could be prime
>6n + 2 = 2(3n + 1) is even
>6n + 3 = 3(2n + 1) is a multiple of 3
>6n + 4 = 2(3n + 2) is even
>6n + 5 could be prime
>6n + 6 = 6(n + 1) is the next multiple of 6.
Therefore each prime (except 2 and 3) is next to a multiple of 6 and that too if that number leaves 1 or 5 as remainder on being divided by 6.
