Why all non prime number are divisible by 2 or 3 or 6 times n plus 1 or 6 times n minus 1?

Answer 1

All non-prime numbers are divisible by prime numbers.

Now the smallest to prime numbers are 2 and 3.

The next Prime number is 5, which is 6*1 - 1.

All larger numbers are in the form of one of 6n, 6n+1, 6n+2, 6n+3, 6n+4, 6n+5.

Now 6n is divisible by 2 and so cannot be a prime.

6n+2 and 6n+4 are also divisible by 2 and so cannot be prime.

6n+3 is divisible by 3 and so cannot be prime.

That leave 6n+1 and 6n+5 as the only two forms than can be prime.

Note though that 6n+5 = 6m-1 where m = n+1.

So all primes are of the form 2, 3, 6n+1 and 6n-1. And all primes can be divided by primes. The result follows.