# Number Divisibility

Important Basic Formulas in number divisibility:

1. n(n + l)(2n + 1) is always divisible by 6.
2. 32n leaves remainder = 1 when divided by 8
3. n3 + (n + 1 )3 + (n + 2 )3 is always divisible by 9
4. 102n + 1 + 1 is always divisible by 11
5. n(n2- 1) is always divisible by 6
6. n2+ n is always even
7. 23n-1 is always divisible by 7
8. 152n-1 +l is always divisible by 16
9. n3 + 2n is always divisible by 3
10. 34n – 4 3n is always divisible by 17
11. n! + 1 is not divisible by any number between 2 and n(where n! = n (n – l)(n – 2)(n – 3)…….3.2.1)
12. For eg 5! = 5.4.3.2.1 = 120 and similarly 10! = 10.9.8…….2.1= 3628800
13. Product of n consecutive numbers is always divisible by n!.
14. If n is a positive integer and p is a prime, then np – n is divisible by p
15. 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 .
16. If n is even , n(n+1)(n+2) is divisible by 24
17. If n is any integer , n^2 + 4 is not divisible by 4
18. 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)