**Important Basic Formulas in number divisibility**:

- n(n + l)(2n + 1) is always divisible by 6.
- 32n leaves remainder = 1 when divided by 8
- n3 + (n + 1 )3 + (n + 2 )3 is always divisible by 9
- 102n + 1 + 1 is always divisible by 11
- n(n2- 1) is always divisible by 6
- n2+ n is always even
- 23n-1 is always divisible by 7
- 152n-1 +l is always divisible by 16
- n3 + 2n is always divisible by 3
- 34n – 4 3n is always divisible by 17
- n! + 1 is not divisible by any number between 2 and n(where n! = n (n – l)(n – 2)(n – 3)…….3.2.1)
- For eg 5! = 5.4.3.2.1 = 120 and similarly 10! = 10.9.8…….2.1= 3628800
- Product of n consecutive numbers is always divisible by n!.
- If n is a positive integer and p is a prime, then np – n is divisible by p
- If a+b+c+d=constant , then the product a^p * b^q * c^r * d^s will be maximum if a/p = b/q = c/r = d/s .
- If n is even , n(n+1)(n+2) is divisible by 24
- If n is any integer , n^2 + 4 is not divisible by 4
- x^n -a^n = (x-a)(x^(n-1) + x^(n-2) + …….+ a^(n-1) ) ……Very useful for finding multiples .For example (17-14=3 will be a multiple of 17^3 – 14^3)