Vedic maths › Dividing by 9
Dividing by 9 with running totals
Because 9 is one less than 10, dividing by 9 needs no times tables: the answer comes from adding the digits as you go along, and the remainder is the total of all the digits.
The method
- The first digit of the answer is the first digit of the number.
- Each next digit of the answer is the one before plus the next digit of the number.
- Stop one digit before the end. The remainder is the last total plus the last digit.
- A total of 10 or more carries 1 to the left.
- A remainder of 9 or more holds a 9: add 1 to the answer for each 9, and keep what is left as the remainder.
Worked examples
2312 ÷ 9
- The answer starts with the first digit.2
- Add the next digit each time, and stop one digit before the end.2 + 3 = 5, 5 + 1 = 6
- The remainder: the last total plus the last digit.6 + 2 = 8
2312 ÷ 9 = 256 remainder 8
1235 ÷ 9
- The answer starts with the first digit.1
- Add the next digit each time, and stop one digit before the end.1 + 2 = 3, 3 + 3 = 6
- The remainder: the last total plus the last digit.6 + 5 = 11
- The remainder holds a 9, so add 1 to the answer.11 = 9 + 2 → 136 + 1 = 137
1235 ÷ 9 = 137 remainder 2
4567 ÷ 9
- The answer starts with the first digit.4
- Add the next digit each time, and stop one digit before the end.4 + 5 = 9, 9 + 6 = 15
- The remainder: the last total plus the last digit.15 + 7 = 22
- A total over 9 carries to the left.4 | 9 | 15: 15: keep 5, carry 1; 9 + 1 = 10: keep 0, carry 1; 4 + 1 = 5 → 505
- The remainder holds 2 nines, so add 2 to the answer.22 = 2 × 9 + 4 → 505 + 2 = 507
4567 ÷ 9 = 507 remainder 4
Why it works
10 = 9 + 1, 100 = 99 + 1 and 1000 = 999 + 1. So 2312 = 2 × 999 + 3 × 99 + 1 × 9 + (2 + 3 + 1 + 2). Divided by 9, that is 2 × 111 + 3 × 11 + 1 × 1 = 222 + 33 + 1 = 256, remainder 2 + 3 + 1 + 2 = 8.
Add 222 + 33 + 1 place by place and you get 2, 2 + 3 and 2 + 3 + 1: the running totals.
Watch out
- This shortcut is for 9 only. Dividing by 8 or 98 uses the general Nikhilam division, which works with the complement (2 for 8) and takes longer.
- Don’t forget the last step: a remainder of 9 or more isn’t finished.
Try these
Work each one out, then tap it to see the answer.
3121 ÷ 9
3, 4, 6, remainder 7: 346 r 7
1022 ÷ 9
1, 1, 3, remainder 5: 113 r 5
2105 ÷ 9
2, 3, 3, remainder 8: 233 r 8
3456 ÷ 9
3, 7, 12 (carry: 382), remainder 18 (2 × 9 + 0): 384 exactly
5214 ÷ 9
5, 7, 8, remainder 12 (9 + 3): 579 r 3
Practise mental maths in the game: play Make Me Smarter, free, with no sign-up.
Source: This method is from Bharati Krishna Tirtha’s Vedic Mathematics (1965). The explanation, examples and questions here are our own.
More Vedic maths
- Squares ending in 5: 75²: 7 × 8 = 56, then 25: 5625.
- All from 9, the last from 10: 1000 − 357: 9 − 3, 9 − 5, 10 − 7: 643.
- Multiplying near 100: 97 × 96: 97 − 4 = 93, 3 × 4 = 12: 9312.
- Squaring near 100: 96²: 96 − 4 = 92, 4² = 16: 9216.
- Vertically and crosswise: 23 × 41: 2 × 4 | 2 × 1 + 3 × 4 | 3 × 1: 943.
- Multiplying by 9, 99, 999: 437 × 999: 436, then 999 − 436 = 563: 436563.
- Last digits adding to 10: 63 × 67: 6 × 7 = 42, 3 × 7 = 21: 4221.
- Multiplying by 11: 253 × 11: 2 | 2 + 5 | 5 + 3 | 3: 2783.
- Checking with digit sums: 437 × 26 = 11362? 5 × 8 → 4, and 11362 → 4: it passes.
- Squaring with the duplex: 43²: 4² | 2 × 4 × 3 | 3² = 16 | 24 | 9: 1849.
Where Vedic maths comes from, all 16 sutras, and our sources: Vedic maths, explained.