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Question Number 130970 by naka3546 last updated on 31/Jan/21

2023^(2021^(2019) ) mod 1000 = x (√((x/9) + ((x +89))^(1/8) )) = y x + y = ?

$$\mathrm{2023}^{\mathrm{2021}^{\mathrm{2019}} } \:\:\mathrm{mod}\:\:\mathrm{1000}\:\:=\:\:{x} \\ $$$$\sqrt{\frac{{x}}{\mathrm{9}}\:+\:\sqrt[{\mathrm{8}}]{{x}\:+\mathrm{89}}}\:\:=\:\:{y} \\ $$$${x}\:+\:{y}\:\:=\:\:\:? \\ $$

Commented by AlagaIbile last updated on 31/Jan/21

Since gcd(2023,1000)=1. λ(1000)= 100 , λ(100)= 20. λ(∙) denote Carmicheal Lambda Function 2023^(2021^(2019 mod 20) mod 100) mod 1000 ≡ 23^(21^(19) mod 100) mod 1000 ≡ 23^(81) mod 1000 ≡ (167)^(27) ≡ (463)^9 ≡ (847)^3 mod 1000 x ≡ 423 mod 1000 ∴ y =(√(((423)/9) + ((423+89))^(1/8) )) = (√(47 + ((512))^(1/8) )) x + y = 423 + (√(47 + 2(2)^(1/8) )) Observation: To make y an integer the question should ask for y = (√((x/9) + ((x + 89))^(1/9) )) = 7 ∴ x + y = 423 + 7 = 450

$$\:{Since}\:\boldsymbol{{gcd}}\left(\mathrm{2023},\mathrm{1000}\right)=\mathrm{1}. \\ $$$$\:\lambda\left(\mathrm{1000}\right)=\:\mathrm{100}\:,\:\lambda\left(\mathrm{100}\right)=\:\mathrm{20}.\:\lambda\left(\centerdot\right)\:{denote} \\ $$$$\:{Carmicheal}\:{Lambda}\:{Function} \\ $$$$\:\mathrm{2023}^{\mathrm{2021}^{\mathrm{2019}\:\:{mod}\:\mathrm{20}} \:{mod}\:\mathrm{100}} \:{mod}\:\mathrm{1000} \\ $$$$\:\equiv\:\mathrm{23}^{\mathrm{21}^{\mathrm{19}} \:{mod}\:\mathrm{100}} {mod}\:\mathrm{1000} \\ $$$$\:\equiv\:\mathrm{23}^{\mathrm{81}} \:\:\:{mod}\:\mathrm{1000} \\ $$$$\:\equiv\:\left(\mathrm{167}\right)^{\mathrm{27}} \:\equiv\:\left(\mathrm{463}\right)^{\mathrm{9}} \:\equiv\:\left(\mathrm{847}\right)^{\mathrm{3}} \:{mod}\:\mathrm{1000} \\ $$$$\:\boldsymbol{{x}}\:\equiv\:\mathrm{423}\:\:{mod}\:\mathrm{1000} \\ $$$$\:\therefore\:\boldsymbol{{y}}\:=\sqrt{\frac{\mathrm{423}}{\mathrm{9}}\:+\:\sqrt[{\mathrm{8}}]{\mathrm{423}+\mathrm{89}}}\:=\:\sqrt{\mathrm{47}\:+\:\sqrt[{\mathrm{8}}]{\mathrm{512}}} \\ $$$$\:\boldsymbol{{x}}\:+\:\boldsymbol{{y}}\:=\:\mathrm{423}\:+\:\sqrt{\mathrm{47}\:+\:\mathrm{2}\sqrt[{\mathrm{8}}]{\mathrm{2}}} \\ $$$$\:\boldsymbol{{Observation}}:\:{To}\:{make}\:\boldsymbol{{y}}\:{an}\:{integer}\:{the}\:{question}\:{should}\:{ask}\:{for} \\ $$$$\:\boldsymbol{{y}}\:=\:\sqrt{\frac{\boldsymbol{{x}}}{\mathrm{9}}\:+\:\sqrt[{\mathrm{9}}]{\boldsymbol{{x}}\:+\:\mathrm{89}}}\:=\:\mathrm{7} \\ $$$$\therefore\:\boldsymbol{{x}}\:+\:\boldsymbol{{y}}\:=\:\mathrm{423}\:+\:\mathrm{7}\:=\:\mathrm{450} \\ $$

Commented by naka3546 last updated on 31/Jan/21

Thank you, sir.

$${Thank}\:{you},\:{sir}. \\ $$

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