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Question Number 128030 by Agnibhoo last updated on 03/Jan/21

99 × 99 = 9801 999 × 999 = 998001 9999 × 9999 = 99980001 99999 × 99999 = ? 999999 × 999999 = ?

$$\:\mathrm{99}\:×\:\mathrm{99}\:=\:\mathrm{9801} \\ $$$$\:\mathrm{999}\:×\:\mathrm{999}\:=\:\mathrm{998001} \\ $$$$\:\mathrm{9999}\:×\:\mathrm{9999}\:=\:\mathrm{99980001} \\ $$$$\:\mathrm{99999}\:×\:\mathrm{99999}\:=\:? \\ $$$$\:\mathrm{999999}\:×\:\mathrm{999999}\:=\:? \\ $$

Answered by Geovanek last updated on 03/Jan/21

99 = 100 − 1 = 10^2 − 1 999 = 1000 − 1 = 10^3 − 1 99...99 (n nines) = 10^n − 1 So, we have (99...99)^2 = (10^n − 1)^2 = 10^(2n) − 2∙10^n + 1 For n = 5, we have 99999^2 = 10^(10) − 2∙10^5 + 1 99999^2 = 10000000000 − 200000 + 1 99999^2 = 9999800001 For n = 6, we have 999999^2 = 999998000001 You put n − 1 “nines” and n − 1 “zeros”.

$$\mathrm{99}\:=\:\mathrm{100}\:−\:\mathrm{1}\:=\:\mathrm{10}^{\mathrm{2}} \:−\:\mathrm{1} \\ $$$$\mathrm{999}\:=\:\mathrm{1000}\:−\:\mathrm{1}\:=\:\mathrm{10}^{\mathrm{3}} \:−\:\mathrm{1} \\ $$$$\mathrm{99}...\mathrm{99}\:\left({n}\:\mathrm{nines}\right)\:=\:\mathrm{10}^{{n}} \:−\:\mathrm{1} \\ $$$$\mathrm{So},\:\mathrm{we}\:\mathrm{have} \\ $$$$\left(\mathrm{99}...\mathrm{99}\right)^{\mathrm{2}} \:=\:\left(\mathrm{10}^{{n}} \:−\:\mathrm{1}\right)^{\mathrm{2}} \\ $$$$=\:\mathrm{10}^{\mathrm{2}{n}} \:−\:\mathrm{2}\centerdot\mathrm{10}^{{n}} \:+\:\mathrm{1} \\ $$$$\mathrm{For}\:{n}\:=\:\mathrm{5},\:\mathrm{we}\:\mathrm{have} \\ $$$$\mathrm{99999}^{\mathrm{2}} \:=\:\mathrm{10}^{\mathrm{10}} \:−\:\mathrm{2}\centerdot\mathrm{10}^{\mathrm{5}} \:+\:\mathrm{1} \\ $$$$\mathrm{99999}^{\mathrm{2}} \:=\:\mathrm{10000000000}\:−\:\mathrm{200000}\:+\:\mathrm{1} \\ $$$$\mathrm{99999}^{\mathrm{2}} \:=\:\mathrm{9999800001} \\ $$$$\mathrm{For}\:{n}\:=\:\mathrm{6},\:\mathrm{we}\:\mathrm{have} \\ $$$$\mathrm{999999}^{\mathrm{2}} \:=\:\mathrm{999998000001} \\ $$$$\mathrm{You}\:\mathrm{put}\:{n}\:−\:\mathrm{1}\:``\mathrm{nines}''\:\mathrm{and}\:{n}\:−\:\mathrm{1}\:``\mathrm{zeros}''. \\ $$

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