The pattern
Take the digits in front of the 5, multiply them by the next whole number, and write 25 at the end. Example: 85^2. 8 x 9 = 72, so the answer is 7225.
Why it works
A number ending in 5 can be written as 10a + 5. Squaring it gives 100a^2 + 100a + 25, which is 100 x a(a + 1) + 25. That is exactly the front part times the next number, followed by 25.
Bonus: same tens digit, ones digits that add to 10
The same idea works when two numbers share a tens digit and their ones digits add to 10. Multiply the tens digit by the next number, then attach the product of the ones digits as two digits. Example: 43 x 47. 4 x 5 = 20, and 3 x 7 = 21. Answer: 2021. Example: 62 x 68. 6 x 7 = 42, and 2 x 8 = 16. Answer: 4216.
Practice problems
- 35^2
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3 x 4 = 12, then attach 25. Answer: 1225.
- 65^2
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6 x 7 = 42, then attach 25. Answer: 4225.
- 105^2
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10 x 11 = 110, then attach 25. Answer: 11025.
References
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