1234 divided by 9.
Done as long division, that is four steps of guess, multiply and subtract, and most children reach for a pen. Done the Vedic way it is three additions you can say out loud: one, three, six, ten. The answer is 137, remainder 1. Here is the method, then the two places it needs care, then why it works.
The method: a running total from the left
Write the number, then keep adding its digits from the left, writing down each total as you go.
1234 ÷ 9
- Bring down the first digit: 1
- Add the next digit: 1 + 2 = 3
- Add the next: 3 + 3 = 6
- Add the last: 6 + 4 = 10. The last total is the remainder, so it goes to one side.
- The answer so far is 136, with 10 left over
- 10 still holds one nine, so take it out: 137 remainder 1
Check: 137 × 9 = 1233, and 1233 + 1 = 1234.
That is the whole method. Every total except the last becomes a digit of the answer; the last total is what is left over. A two-digit number shows it at its smallest: 23 ÷ 9 is 2, then 2 + 3 = 5, so 2 remainder 5.
Care point one: a remainder of 9 or more
A remainder can never be 9 or more, so when the last total reaches 9, take out every nine it holds and add them to the answer.
2016 ÷ 9
- Totals: 2, 2 + 0 = 2, 2 + 1 = 3, and last 3 + 6 = 9
- Answer so far 223, with 9 left over
- 9 is exactly one nine: 224, remainder 0
The left-over total can hold more than one nine. In 8888 ÷ 9 the totals are 8, 16, 24 and 32: 32 holds three nines with 5 over, so whatever the first three totals give, add 3 to it and the remainder is 5. That first part needs the second care point.
Care point two: a total that passes 9
When a running total goes into two digits, it no longer fits in its column. Keep the units digit there and carry the tens to the column on its left, exactly as in ordinary addition.
7543 ÷ 9
- Totals: 7, 7 + 5 = 12, 12 + 4 = 16, and last 16 + 3 = 19
- The answer columns hold 7 | 12 | 16. Start from the right: keep the 6, carry 1 onto the 12 to make 13
- Keep the 3, carry 1 onto the 7 to make 8
- Answer so far 836, with 19 left over
- 19 is two nines and 1: 838, remainder 1
Check: 838 × 9 = 7542, and 7542 + 1 = 7543.
Back to 8888: the columns 8 | 16 | 24 settle to 984 (keep 4, carry 2 onto 16 to make 18; keep 8, carry 1 onto 8 to make 9). Add the three nines from the remainder and the answer is 987 remainder 5. Children who find the carrying awkward can leave it until the plain version is automatic; numbers with small digits never need it.
Why it works
Every ten is a nine with one left over. That one fact is the whole reason.
1234, piece by piece
- 1000 is 111 nines, with 1 over
- 200 is 22 nines, with 2 over
- 30 is 3 nines, with 3 over
- 4 is no nines, with 4 over
- Nines: 111 + 22 + 3 = 136. Left over: 1 + 2 + 3 + 4 = 10
Look at the two totals. 136 is exactly what the running totals 1, 3, 6 spell out, and 10 is exactly the last running total. The method is just this sum, done without writing the pieces down.
It also explains something the digit-sum check relies on: the remainder when you divide by 9 is the digit sum of the number, with nines taken out. 1 + 2 + 3 + 4 = 10, which leaves 1, and 1234 divided by 9 leaves 1.
The same idea for 8
Dividing by 9 is the simplest case of Nikhilam division, the division method that comes from the sutra Nikhilaṁ Navataścaramaṁ Daśataḥ, "all from nine and the last from ten", the same sutra behind the near-100 multiplication. 9 is 1 short of 10, so each new total adds the previous total once. 8 is 2 short of 10, so each new total adds the previous total twice.
1234 ÷ 8
- Bring down the first digit: 1
- 2 + (2 × 1) = 4
- 3 + (2 × 4) = 11
- Last: 4 + (2 × 11) = 26, the remainder part
- Columns 1 | 4 | 11 settle to 151; 26 is three eights and 2, so 154 remainder 2
Check: 154 × 8 = 1232, and 1232 + 2 = 1234.
The further the divisor is from 10, the faster the totals grow, which is why the method is used for divisors just below 10, 100 or 1000 and not for, say, 6. For a number like 7 or 13, ordinary division is quicker.
Five to try
| Question | Running totals | Answer |
|---|---|---|
| 52 ÷ 9 | 5, 7 | 5 remainder 7 |
| 341 ÷ 9 | 3, 7, 8 | 37 remainder 8 |
| 4005 ÷ 9 | 4, 4, 4, 9 | 445 remainder 0 |
| 6142 ÷ 9 | 6, 7, 11, 13 | 682 remainder 4 |
| 2025 ÷ 8 | 2, 4, 10, 25 | 253 remainder 1 |
Cover the last two columns and work each one aloud. 6142 needs both care points: the columns 6 | 7 | 11 carry to 681, and the 13 left over holds one nine, so 682 remainder 4.
Where this fits
Division comes after the multiplication methods in the order we teach the sutras in, and dividing by 9 is the right first step into it: no guessing, no subtraction, and an answer the child can check by multiplying back. For multiplying by 9, 99 and 999, see its own post.
Learning the rest of the sutras
Ms. Ishita Parikh holds the International Vedic Maths Teachers' Training certification at Grade A, a six-month programme approved under Skill India and the NSDC. Classes cover all sixteen sutras with the reasoning underneath each one, taught to children as methods they can explain, not tricks they have to trust.
See how the course is structured, or book the free first class and try these live.
Questions people ask
How do you divide by 9 using Vedic Maths?
Keep a running total of the digits from the left. Each total before the last digit is a digit of the answer, and the final total is the remainder. For 1234 divided by 9 the totals are 1, 3, 6 and 10, so the answer is 136 with 10 left over; 10 holds one more nine, which makes it 137 remainder 1.
Why does the divide-by-9 trick work?
Because every ten is a nine with one left over. In 1234, the thousand is 111 nines and 1 over, the 200 is 22 nines and 2 over, the 30 is 3 nines and 3 over, and the 4 is all left over. That is 111 + 22 + 3 = 136 nines, and 1 + 2 + 3 + 4 = 10 left over, which holds one more nine: 137 remainder 1.
What if the remainder comes out as 9 or more?
Take out as many nines as it holds and add that many to the answer. In 2016 divided by 9 the remainder comes out as exactly 9, so the answer goes from 223 to 224 with nothing left over. In 8888 divided by 9 it comes out as 32, which is three nines and 5, so 984 becomes 987 remainder 5.
Can the same method divide by 8?
Yes. 8 is 2 short of 10, so instead of adding the previous total you add twice the previous total. For 1234 divided by 8 the totals are 1, then 2 + 2 = 4, then 3 + 8 = 11, then 4 + 22 = 26. That gives 151 with 26 left over, and 26 is three eights and 2, so the answer is 154 remainder 2.
What age can a child learn to divide by 9 this way?
Any child who can add single digits in their head and knows what a remainder is. The carrying step needs a little more confidence, so it is best taught once the plain version is automatic.