### Vedic Math - Divsion by 9

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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.