**Basic Formulae of Divisibility from 2 to 19:**

**1. Divisibility by 2 :** If the last digit of a number is 0 or an even number then that number is divisible by 2.

Such as, 242, 540 etc.

**2. Divisibility by 3 :** If the sum of all digits of a number is divisible by 3, then that number will be divisible by 3. Such as.

432 : 4 + 3 + 2 = 9 which is divisible by 3.

So, 432 is divisible by 3.

**3. Divisibility by 4 :** If in any number last two digits are divisible by 4, then whole number will be divisible by 4. Such as, 48424. In this number 24 is divisible by 4. So, 48424 will be divisible by 4.

**4. Divisibility by 5 :** If last digit of a number is 5 or 0, then that number is divisible by 5. Such as 200, 225 etc. 5. Divisibility by 6 : If a number is divisible by both 2 and 3, then that number is divisible by 6 also, such as 216, 25614 etc.

**5. Divisibility by 6 :** If a number is divisible by both 2 and 3, then that number is divisible by 6 also, such as 216, 25614 etc.

**6. Divisibility by 7 :** Here concept of oscillator should be applied. The meaning of negative oscillator is – there increases or decreases 1 from the factor of 10 of the number.

As, 21 : 2 × 10 + 1 = 21

49 : 5 × 10 – 1 = 50 – 1 = 49

To check the divisibility of 7, we use oscillator ‘2’, as ,

112 : 11 – 2 × 2 = 7 which is divisible by 7

Again,

343 : 34 – 2 × 3 = 28 which is divisible by 7. Then 343

will be divisible by 7.

**7. Divisibility by 8 :** If in any number last three digits are divisible by 8, then whole number is divisible by 8, such as, 247864 since 864 is divisible by 8.

So, 247864 is divisible by 8.

Similarly, 289000 is divisible by 8.

**8. Divisibility by 9 :** If the sum of all digits of a number is divisible by 9, then that whole number will be divisible by 9. As, 243243 : 2 + 4 + 3 + 2 + 4 + 3 = 18 is divisible by 9.

So, 243243 is divisible by 9.

**9. Divisibility by 10 :** The number whose last digit is ‘0’, is divisible by 10, such as, 10, 20, 200, 300 etc.

**10. Divisibility by 11 :** If the difference between “Sum of digits at even place” and “Sum of digits at odd place” is divisible by 11, then the whole number is divisible by 11. such as,

∴ (9 + 7) – (4 + 1) = 16 – 5 = 11 is divisible by 11.

So, 9174 will be divisible by 11.

**11. Divisibility by 12 :** If a number is divisible by 3 and 4 both. Then the number is divisible by 12. Such as, 19044 etc.

**12. Divisibility by 13 :** For 13 we use oscillator 4, but our oscillator is not negative here. It is one-more oscillator (4).

143 : 14 + 3 × 4 = 26

and 26 is divisible by 13, So, 143 is divisible by 13.

Similarly for 325 : 32 + 5 × 4 = 52

52 is divisible by 13

Hence, 325 will also be divisible by 13.

**13. Divisibility by 14 :** If a number is divisible by 2 and 7 both then that number is divisible by 14 i.e. number is even and oscillator 2 is applicable.

**14. Divisibility by 15 :** If a number is divisible by 3 and 5 both, then that number is divisible by 15.

**15. Divisibility by 16 :** If last 4 digits of a number are divisible by 16, then whole number is divisible by 16. Such as 341920.

**16. Divisibility by 17 :** For 17, there is a negative ‘oscillator 5’. This process is same as the process of 7. As.

1904 : 190 – 5 × 4 = 170.

∵ 170 is divisible by 17. So 1904 will be divisible by 17.

**17. Divisibility by 18 :** If a number is divisible by 2 and 9 both, then that number is divisible by 18.

**18. Divisibility by 19 :** For 19, there is one–more (positive) oscillator 2, which is same processed as 13. As,

361 = 36 + 1 × 2 = 38

∵ 38 is divisible by 19. So 361 is also divisible by 19.

Few more Important Points:

1. Out of a group of n consecutive integers one and only one number is divisible by n.

2. The product of n consecutive numbers is always divisible by n! or = n.

3. For any number n, (n^{p}–h) is always divisible by P where P is a prime number, for e.g.,

if n = 2 and P = 5 then,

(2^{5} – 2) = (32 – 2) = 30 which is divisible by 5.

4. The square of an odd number when divided by 8 always leaves a remainder 1. as If we divide 7^{2} = 49 or 5^{2} = 25 by 8 then remainder will be 1.

5. For any natural number n, n^{5} or n^{4k + 1} is having same unit digit as n has, where k is a whole number. such as,

3^{5} = 243 has 3 at its unit place.

6. Square of any natural number can be written in the form of 3n or 3n + 1 or 4n or (4n + 1).

e.g. square of 11 = 121 = 3 × 40 + 1

or 4 × 30 + 1

If N = a^{p} b^{q} c^{r} ……… where a, b and c are prime numbers and p, q and r are natural numbers. then

1. Number of factors of N is given by

F = (p + 1)(q + 1)(r + 1) …….

2. Number of ways to express the number as a product of two factors are

F/2 if F is even or (F+1)/2 if F is odd respectively.

3. Sum of all the factors of the number N.

4. The number of ways in which a number N can be resolved into co–prime factors is 2^{k – 1}, where k is the number of different Prime factors of the number N.

5. The number of co–primes to number N is given by

Special Rules :

Rule 1 : If the sum of digits of two digit number is ‘a’ and if the digits or the number are reversed.. such that number reduces by ‘b’, then

Original Number = (11a + b ) / 2

For example : (For number 82) a = 8 + 2 = 10.

and b = 82 – 28 = 54 is given then

original number = (11 × 10 + 54) / 2 = 164/2 = 82

Rule 2 : If the sum of digits of two digit number is ‘a’ and if the digits of the number are reversed, such that number increases by ‘b’. then,

Original Number = (11a – b)/2

e.g. (For number 47): a = 4 + 7 = 11.

& b = 74 – 47 = 27 thus the

original number = (11 × 11 – 27) / 2 = 47.

Rule 3 : If the difference between a number and formed by number reversing digit is x, then the difference between both the digits of the number is x/9.

eg. (for 63) x = 63 – 36 = 27

Required difference = 27 / 9 = 3

Rule 4 : If the sum of a number and the number formed by reversing the digits is x, then the sum of digits of the number is x / 11.

e.g. (For number 76) = x = 67 + 76 = 143.

Required sum of numbers = 67 + 76 = 143.

Required sum = 143 / 11 = 13

Dividend = (Divisor × Quotient) + Remainder.

Divisor = (Dividend – Remainder) / Quotient.

Quotient = (Dividend – Remainder) / Divisor.

Remainder = Dividend – (Divisor × Quotient).

**Special Rule for Remainder Calculation:**

Rule 5 : If a^{n} / a-1 then remainder will always be 1, whether n is even or odd.

Rule 6 : If a^{(even number)} / (a+1) , then remainder will be 1.

Rule 7 : If a^{(odd number)} / (a+1) , then remainder will be a.

Rule 8 : If n is a single digit number, then in n^{3}, n will be at unit place. It is valid for the number 0, 1, 4, 5, 6 or 9. As, digit at unit place in (4^{3}) is 4.

Rule 9 : If n is a single digit number then in np, where p is any number (+ve), n will be at unit place. It is valid for 5 and 6.

Follow