Square 497 in your head

Question

Solve in your head. What is 497^2? Aim: under 10 seconds.

Accepted Answer

Use the slide n^2 = (n-d)(n+d) + d^2 with d = 3, so one factor becomes 500: 497^2 = (494)(500) + 9 = 247{,}000 + 9 = 247{,}009. Multiplying by 500 is easy: halve 494 to get 247, then append three zeros for the thousands, giving 247{,}000, and add 3^2 = 9. Cross-check with the binomial expansion around 500: 497^2 = (500 - 3)^2 = 250{,}000 - 3{,}000 + 9 = 247{,}009. \checkmark Both routes agree, but the slide form has no middle term to fumble. Takeaway. Near a round multiple, pick d so one factor lands on that round number: 497^2 = 494 	imes 500 + 9. Multiplying by 500 is just "halve, then add…

Square 497 in your head

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