Vedic maths › Last digits adding to 10
Multiplying when the last digits add up to 10
The sub-sutra Antyayordaśake’pi, “when the last digits add up to ten”, covers pairs like 63 × 67 or 112 × 118: the same front, and last digits that add up to 10. It is the squares ending in 5 method made general: 75 × 75 is the pair where both last digits are 5.
The method
- Check the pair: the same front (6 and 6) and last digits that add up to 10 (3 + 7).
- Left part: the front times one more than itself: 6 × 7 = 42.
- Right part: multiply the last digits, and write two digits: 3 × 7 = 21, and 1 × 9 = 09.
- Put them side by side: 4221.
Worked examples
63 × 67
- Both start with 6, and the last digits add up to 10.3 + 7 = 10
- Left part: 6 times one more than itself.6 × 7 = 42
- Right part: multiply the last digits, and write two digits.3 × 7 = 21
63 × 67 = 42 | 21 = 4221
81 × 89
- Both start with 8, and the last digits add up to 10.1 + 9 = 10
- Left part: 8 times one more than itself.8 × 9 = 72
- Right part: multiply the last digits, and write two digits.1 × 9 = 09
81 × 89 = 72 | 09 = 7209
112 × 118
- Both start with 11, and the last digits add up to 10.2 + 8 = 10
- Left part: 11 times one more than itself.11 × 12 = 132
- Right part: multiply the last digits, and write two digits.2 × 8 = 16
112 × 118 = 132 | 16 = 13216
Why it works
Write the numbers as 10a + b and 10a + c, with b + c = 10. Then (10a + b)(10a + c) = 100a² + 10a(b + c) + bc = 100a² + 100a + bc = 100 × a(a + 1) + bc.
Watch out
- The fronts must be the same: 63 × 77 doesn’t work this way.
- Always write the right part with two digits: 81 × 89 is 72 | 09, not 72 | 9.
- It stretches to the last two digits adding up to 100, with four digits on the right: 192 × 108 is 1 × 2 = 2, then 92 × 8 = 0736: 20736.
Try these
Work each one out, then tap it to see the answer.
44 × 46
4 × 5 = 20, 4 × 6 = 24: 2024
72 × 78
7 × 8 = 56, 2 × 8 = 16: 5616
97 × 93
9 × 10 = 90, 7 × 3 = 21: 9021
31 × 39
3 × 4 = 12, 1 × 9 = 09: 1209
124 × 126
12 × 13 = 156, 4 × 6 = 24: 15624
Practise it in the game: Speed Tricks, 2.4 Units add up to 10. 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.
- 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.
- 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.