Numbers Concepts -1

1. For any positive integer n, product of ‘n’ or more than ‘n’ consecutive positive integers is divisible by n!.

For example, 63*64*65*.......*91 is divisible by 29!
2. If (n-1)! is not divisible by ‘n’, then n is a prime number. 4 is a special case here, which obeys this rule but not a prime.
3. From 1 & 2 above, we can conclude that,  for any positive integer 'n-1', if the product of ‘n-1’  consecutive positive integers is not divisible by n, then n is a prime. But if divisible, we can't say that n is not a prime.
4. If a > b ≥ 3, then b^a > a^b where a,b Є Z+
5. Product of two successive integers always ends in 2,6, or 0
6. The general perception about even numbers is even numbers start with 0 and go on 2,4,6,8,... and the odd numbers are 1,3,5,.... But negative integers are also to be categorized in to even and odd sets.

Hence the even number set is: ....-4,-2,0,2,4,6,.....
and the odd number set is: ....-5,-3,-1,3,5,7,.....
7. Why '1' is not a prime number?
A prime number can be defined as a positive integer that has exactly two different positive divisors, 1 and itself.  As the number '1' has only one positive divisor (ie., itself), it is not a prime number.