### Vedic Math - Divsion by 9

Let us learn, how to do 'Division' operation using 'Vedic Math'. Conventionally, we do it like following:

Divisor ) Dividend ( Quotient
---------
---------
_________
Remainder

However, in the Vedic process, the format is
Divisor ) Dividend
--------
__________________
Quotient | Remainder

Let us first start with one of the special case of division i.e. Division By 9, a very interesting and simple technique.

When dividing by 9, the remainder is always the digit sum of the original number.

For 2-digit number divided by 9
To divide ab by 9 : Rewrite ab as a | b . The quotient is a, and the remainder is simply a + b.

a  |  b
|  a
---------
a  | a + b

Examples:

12 divided by 9
Here quotient = 1 and remainder = 1+2 = 3

23 divided by 9
Here quotient = 2 and remainder = 2+3 = 5

70 divided by 9
Here quotient = 7 and remainder = 7+0 = 7

Now, let us discuss the cases when remainder is greater than 9 :-

86 divided by 9
Here quotient = 8 and remainder= 8+6 = 14 ( >9 )
So we add one in the quotient and becomes 9 ; and
remainder becomes 5, after subtracting 9 from 14

New quotient =  9 and New remainder = 5

75 divided by 9
Here quotient = 7 and remainder = 7+5 = 12 ( >9 )
So, New quotient =  8 and New remainder = 3 (12-9=3)

Also, notice here, that the new remainder is just the digit sum of the old remainder.

For 3-digit number divided by 9

ab  |  c
a  | a + b
---------------
ab + a | a + b + c

Quotient: ab + a ; Remainder: a + b + c. However, remember that the remainder should be less than 9. And if remainder is greater than 9; we add 1 to quotient and subtract 9 from the remainder.

Examples:
124 divided by 9
12 |  4
1 | 1 + 2
------------
13 | 7
Quotient = 13 and Remainder = 7

311 divided by 9
31 |  1
3 | 3 + 1
------------
34 | 5
Quotient = 34 and Remainder = 5

267 divided by 9
26 |  7
2 | 2 + 6
------------
28 | 15       (add 1 to quotient ; subtract 9 from remainder or digit sum of the remainder i.e. 1+5=6)
29 | 6
Quotient = 29 and Remainder = 6

Examples for 4-digit number
3121 divided by 9

3172 divided by 9

Example for 5-digit number
42111 divided by 9

Example for 6-digit number
214091 divided by 9

21409  | 1
The first digit 2 is write down as the first digit of the quotient. Take this 2 and add to the next digit '1'. This gives 3 as the next
digit. Working this way 3+4 =7, 7+0 =7 , 7+9 = 16 and the remainder is 16+1 = 17
2377 16 | 17
carry 1 on the left, gives
23786 | 17
The remainder 17 >  9 , so add 1 to quotient and subtract 9 from remainder.
23787 | 8
Q= 23787, R = 8

I hope, you would enjoy using this interesting and simple technique. In case of any query, please post in the comments.