The document discusses the Nikhilam method for division. It provides examples and explanations for dividing numbers with divisors of 9, 8, 7 and 6. There are four cases covered: 1) divisor of 9 with 2-digit or more dividends, 2) divisor of 9 with 3+ digit dividends, 3) divisor of 9 with remainders greater than 9, and 4) divisors of 8, 7, 6. The method involves placing digits below a line, adding adjacent digits, and using remainders to get the quotient and remainder. Worked examples with step-by-step explanations are provided for each case.

1. Division - Nikhilam Sutra
Vedic Mathematics
Composed by A V Prakasam for the benefit of his daughters
from the book ‘Vedic Mathematics’
By Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja
DIVISION – Nikhilam Sutra

2. Division - Nikhilam Sutra (Contd.)
Division - By the Nikhilam Method
Special Cases:
Case 1: Divisor 9, Dividend 2 digits.
Example: 12/9 9) 1 2
1 Add
Quotient 1 3 Reminder
Explanation:
- The divisor contains one digit. So, put a bar before one digit from the right of the dividend.
- Next draw a line below the dividend, after leaving a line for entry of figures
- Bring the first digit of the dividend below the line
- Add this digit to the second digit of the dividend and put it below the line after the first digit.
- The first digit below the line gives the quotient and the second digit the reminder.
Notice that the reminder is equal to the some of the digits of the quotient
Practice with the following exercises:
(i). 21/9 (ii) 33/9 (iii). 40/9 (iv) 52/9 (v) 61/9 (vi) 70/9 (vii) 80/9
Note that we have chosen the dividends which have a reminder
less than nine only, as a first step.

3. Division - Nikhilam Sutra (Contd.)
Case 2: Divisor 9, Dividend 3 digits or more, Reminder less
than 9. Add Add
Example: (i) 103/9 9) 1 0 3
1 1
Quotient 1 1 4 Reminder
(ii) 12031/9 9) 1203 1
133 6
Quotient 1336 7 Reminder
Explanation:
- The divisor contains one digit. So, put a bar before one digit from the right of the dividend.
- Next draw a line below the dividend, after leaving a line for entry of figures
- Bring the first digit of the dividend below the line
- Add this digit to the second digit of the dividend and put the sum below the line after first digit.
- Add this second digit below the line to the third digit and put the sum below the line
- Continue this process till all he digits of the dividend are covered.
- The digits before the bar line gives the quotient and the digit after the bar line the reminder.
Notice that the reminder in these cases also is equal to the some of the digits of the
quotient
Practice with the following exercises:
(i). 113/9 (ii) 124/9 (iii). 311/9 (iv) 12301/9 (v) 120021/9
Note that we have chosen the dividends which have a reminder less than nine only, in these
cases also

4. Division - Nikhilam Sutra (Contd.)
Case 3: Divisor 9, Dividend 2 or more digits, Reminder greater than or
equal to 9
Example: (i) 18/9 9) 1 8
1
1 9 First Reminder
1 0
Quotient 2 0 Final Reminder
(ii) 136/9 9) 13 6
1 4
1 4 10 First Reminder
1 1
Quotient 15 1 Final Reminder
Explanation :
- After carrying out the division as in the previous case, the reminder is 9 or more
- Carry out a further division of this reminder to arrive at a quotient and the final reminder.
- Add this quotient to the first quotient
- The sum of the quotients and the final reminder give the answer.
- Note that the second division can be carried out using the same ‘Nikhilam’ method or by direct division, if it is
simple.
Practice with the following exercises:

5. Division - Nikhilam Sutra (Contd.)
Case 4: Divisors are 8,7,6 etc.
Examples: (i) 23/8 8) 2 3
4 (2*2)
2 7
(ii) 12/7 7) 1 2
3 (1*3)
1 5
(iii) 11/6 6) 1 1
4 (1*4)
1 5
Explanation:
- Take the 10’s complement of the divisor. Call it the ‘divisor complement’
- It is 2 in Example (i), 3 in example (ii) and 4 in example (iii)
- Bring the first digit of the dividend below the horizontal line. Call it the ‘quotient digit’.
- Multiply the quotient digit by the divisor complement.
- Place the product below the second digit, above the horizontal line and add.
- The first digit below the line is the quotient and the second digit is the reminder
Practice with the following exercises :
(i) 31/8 (ii) 15/7 (ii) 17/6
Note that we have chosen the dividends which have a reminder less than the divisor. If the
Reminder is greater than the divisor, proceed as in case 3.

6. Division - Nikhilam Sutra (Contd.)
Examples of divisions with bigger numbers :
In the following examples, the compliments of the divisor are shown under
the line below the divisor. E.g., 100’s complement of 89 is 11, 1000’s complement
of 888 is 112, 10000’s complement of 7999 is 2001 etc.
(1) 111/89 89) 1 11 (2) 1234/888 888) 1 234 (3) 12345/7999 7999) 1 2345
11 11 112 112 2001 2001
1 22 1 346 1 4346
(4) 1111/89 89) 11 11 (5) 12345/888 888) 12 345 (6) 210012/8997 8997) 21 0012
11 1 1 112 1 12 1003 2 006
22 336 3009
12 43 13 7911 23 30711
13 7101 23 3081
13 801
(7) 11111111/99979 99979) 111 11111 (8) 11001/88 88) 110 01
00021 00 021 12 12
0 0021 24
00021 48
111 13442 124 89
=1251

7. Division - Nikhilam Sutra (Contd.)
Examples of divisions with bigger numbers :
In the following examples, the compliments of the divisor are shown under
the line below the divisor. E.g., 100’s complement of 89 is 11, 1000’s complement
of 888 is 112, 10000’s complement of 7999 is 2001 etc.
(1) 111/89 89) 1 11 (2) 1234/888 888) 1 234 (3) 12345/7999 7999) 1 2345
11 11 112 112 2001 2001
1 22 1 346 1 4346
(4) 1111/89 89) 11 11 (5) 12345/888 888) 12 345 (6) 210012/8997 8997) 21 0012
11 1 1 112 1 12 1003 2 006
22 336 3009
12 43 13 7911 23 30711
13 7101 23 3081
13 801
(7) 11111111/99979 99979) 111 11111 (8) 11001/88 88) 110 01
00021 00 021 12 12
0 0021 24
00021 48
111 13442 124 89
=1251