Vedic maths › Squares ending in 5
Squaring numbers that end in 5
Squaring a number that ends in 5 takes one small multiplication. Cover the 5, multiply what is left by one more than itself, and write 25 after it. The sutra behind it, Ekādhikena Pūrveṇa, says just that: “by one more than the one before”.
The method
- Cover the 5 and look at the number in front of it: 7 for 75, 11 for 115.
- Multiply that number by the next one up: 7 × 8 = 56.
- Write 25 after the result: 5625.
Worked examples
65²
- Cover the 5: 6 is left. One more than 6 is 7.6 × 7 = 42
- Write 25 after it.42 | 25
65² = 4225
95²
- Cover the 5: 9 is left. One more than 9 is 10.9 × 10 = 90
- Write 25 after it.90 | 25
95² = 9025
125²
- Cover the 5: 12 is left. One more than 12 is 13.12 × 13 = 156
- Write 25 after it.156 | 25
125² = 15625
Why it works
A number that ends in 5 is 10a + 5, where a is the part in front of the 5. Its square is (10a + 5)² = 100a² + 100a + 25 = 100 × a(a + 1) + 25. So the hundreds are a × (a + 1), and the last two digits are always 25.
Watch out
- It only works for numbers that end in 5. For others, try squaring near a base or the duplex.
- Write the 25 even when the product ends in 0: 105² is 10 × 11 = 110, then 25: 11025.
- Bigger numbers just need a bigger multiplication: 245² is 24 × 25 = 600, then 25: 60025.
- It works with a decimal point too: 7.5² = 56.25, because 7 × 8 = 56 and 0.5 × 0.5 = 0.25.
Try these
Work each one out, then tap it to see the answer.
45²
4 × 5 = 20, then 25: 2025
55²
5 × 6 = 30, then 25: 3025
85²
8 × 9 = 72, then 25: 7225
105²
10 × 11 = 110, then 25: 11025
145²
14 × 15 = 210, then 25: 21025
Practise it in the game: Speed Tricks, 2.3 Squares ending in 5. 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
- 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.
- 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.