NUMBERS - Important Formulae/Notes

(1) No. of Factors / Co-primes

If any integer is be expressed in the below format,


then,


Where,
Nf → No. of Factors of N
Sf → Sum of all the Factors of N
Nc → No. of co-primes to N (that are lesser than N)
Sc → Sum of all co-primes to N (that are lesser than N)


(2) Divisibility Rule

3 - Sum of the digits should be divisible by 3
Eg: 36 → 3+6=9 (divisible by 3)

4 - Last 2 digits of the number should be divisible by 4
Eg: 2248 → last 2 digits = 48 (divisible by 4) → 2248 is divisible by 4

6 - Number should be divisible by both 2 & 3
Eg; 5124 is divisible by 2. Also 5+1+2+4=12 (divisible by 3) → 5124 is divisible by 6

7 - If Number is of the form (N = abc), then (ab - 2c) should be divisible by 7
Eg: 322 → 32-2(2)=28 (divisible by 7) → 322 is divisible by 7

8 - Last 3 digits of the number should be divisible by 8
Eg: 36184 → Last 3 digits = 184 (divisible by 8) → 36184 is divisible by 8

9 - Sum of the digits should be divisible by 9
Eg: 4302 → 4+3+0+2=9 (divisible by9) → 4302 is is divisible by 9

11 - (sum of even digits) - (sum of odd digits) should be a multiple of 11
Eg: 6182 → (6+8)-(1+2)=11 (divisible by 11) → 6182 is divisible by 11

19 - If Number is of the form (N = abc), then (ab + 2c) should be divisible by 19
Eg: 798 → 79+2(8)=95 (19*5=95, divisible by 19) → 798 is divisible by 19