Vedic maths › Checking with digit sums

Checking answers with digit sums

A digit sum check tells you in a few seconds whether a multiplication is wrong. It is much older than Vedic maths: Indian mathematicians checked their arithmetic this way by the 10th century, Ibn Sīnā called it the Hindu method, and it reached Europe through Fibonacci’s Liber Abaci in 1202. Tirthaji’s book doesn’t teach it, but Vedic maths courses do, as the way to check every answer.

The method

1. Find each number’s digit sum: add its digits, and keep adding until one digit is left. 437: 4 + 3 + 7 = 14, then 1 + 4 = 5.
2. Multiply the two digit sums, and take the digit sum of that: 5 × 8 = 40, which gives 4.
3. Find the digit sum of the answer: 11362 gives 13, then 4.
4. If they differ, the answer is wrong. If they match, it is very likely right.
5. A 9 counts the same as 0, so you can skip 9s, and digits that add up to 9, as you go: that is casting out nines.

Worked examples

Why it works

10 is one more than 9, so 10, 100, 1000 and every other power of 10 leaves 1 over when divided by 9. So a number leaves the same amount over as its digit sum: 437 = 4 × 100 + 3 × 10 + 7 leaves what 4 + 3 + 7 leaves.

What is left over multiplies along with the numbers: if a leaves 5 and b leaves 8, then a × b leaves what 5 × 8 leaves. A right answer must agree.

The sutra’s own use is the same idea in algebra: (x + 2)(x + 5) = x² + 7x + 10, and the sums of the coefficients agree, 3 × 6 = 18 = 1 + 7 + 10. Adding up coefficients is putting 1 in place of x; a digit sum puts 1 in place of 10, which is what 10 leaves over after taking out 9.

Watch out

• It misses mistakes that change the answer by a multiple of 9: two swapped digits (11362 and 11632) or a lost 0 give the same digit sum.
• It works for adding and subtracting too: the digit sums add or subtract the same way.

Try these

Work each one out, then tap it to see the answer.

1. Is 34 × 27 = 918?

   7 × 9 → 9, and 918 → 9: yes, it passes.

2. Is 56 × 43 = 2418?

   2 × 7 → 5, but 2418 → 6: no. It is 2408.

3. Is 125 × 32 = 4000?

   8 × 5 → 4, and 4000 → 4: yes, it passes.

4. Is 73 × 48 = 3504?

   1 × 3 → 3, and 3504 → 3: yes, it passes.

5. Is 89 × 64 = 5796?

   8 × 1 → 8, but 5796 → 9: no. It is 5696.

Practise it in the game: Speed Tricks, 3.2 Check with nines. Free, with no sign-up.

Source: Casting out nines is older than Vedic maths, and Tirthaji’s book checks algebra with this sutra rather than sums with digit sums; Vedic maths courses teach both. 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.
• Dividing by 9: 2312 ÷ 9: 2, 2 + 3, 5 + 1: 256, remainder 6 + 2 = 8.
• 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.