Square 305 with the ends-in-5 shortcut

Question

The ends-in-5 rule doesn't care how many digits are in front. Solve in your head. What is 305^2? Aim: under 12 seconds.

Accepted Answer

Strip the trailing 5: the leading number is 30. Multiply it by the next integer, 31, then append 25: 30 	imes 31 = 930 \ \Rightarrow\ 305^2 = 93025. So 305^2 = \mathbf{93025}. The front is 930, the back is 25. Why: (10n+5)^2 = 100\,n(n+1) + 25 with n = 30, giving 100 	imes 930 + 25 = 93025. Expansion check: (300+5)^2 = 90000 + 3000 + 25 = 93025. Matches. Takeaway. Ends-in-5 works at any size: multiply the "everything-before-the-5" number by one more than itself, then append 25.

Square 305 with the ends-in-5 shortcut

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