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Question Number 126677 by deleteduser1 last updated on 26/Sep/22

Commented by talminator2856791 last updated on 23/Dec/20

$$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{question}?$$$$$$

Commented by talminator2856791 last updated on 23/Dec/20

$$\mathrm{what}\mathrm{is}\mathrm{mod}\mathrm{in}\mathrm{this}\mathrm{context}$$

Commented by talminator2856791 last updated on 23/Dec/20

$$$$$$\mathrm{ooooooooooh}$$

Answered by floor(10²Eta[1]) last updated on 24/Dec/20

$${2020}^2=2^4.5^2.{101}^2$$$$-{\left(2^{2020}\right)}^5\equiv 0\left({\mathrm{mod}2}^4\right)$$$$\varphi \left(5^2\right)=20\Rightarrow -{\left(2^{505}\right)}^{20}\equiv -1\left({\mathrm{mod}5}^2\right)$$$$\varphi \left({101}^2\right)={101}^2-101=10100$$$$\Rightarrow -{\left(2^{2020}\right)}^5=-2^{10100}\equiv -1\left({\mathrm{mod}101}^2\right)$$$$\mathrm{let}\mathrm{A}=-{\left(2^{2020}\right)}^5:$$$$\mathrm{A}\equiv 0\left(\mathrm{mod}{2}^4\right)$$$$\mathrm{A}\equiv -1\left(\mathrm{mod}{5}^2\right)$$$$\mathrm{A}\equiv -1\left(\mathrm{mod}{101}^2\right)$$$$\mathrm{A}={101}^2\mathrm{x}-1$$$${101}^2\mathrm{x}-1\equiv -1\left({\mathrm{mod}5}^2\right)$$$$\Rightarrow \mathrm{x}\equiv 0\left({\mathrm{mod}5}^2\right)\Rightarrow \mathrm{x}=5^2\mathrm{y}$$$$\Rightarrow \mathrm{A}={101}^2.5^2\mathrm{y}-1$$$${101}^2{5}^2\mathrm{y}-1\equiv 0\left({\mathrm{mod}2}^4\right)$$$$\equiv 9^2\mathrm{y}-1\equiv \mathrm{y}-1\equiv 0\left({\mathrm{mod}2}^4\right)$$$$\mathrm{y}\equiv 1\left({\mathrm{mod}2}^4\right)\Rightarrow \mathrm{y}=2^4\mathrm{z}+1$$$$\Rightarrow \mathrm{A}={101}^2.5^2(2^4\mathrm{z}+1)-1$$$$={101}^2.5^2.2^4\mathrm{z}+{101}^2.5^2-1$$$$={2020}^2\mathrm{z}+255024\equiv 255024\left({\mathrm{mod}2020}^2\right)$$$$$$

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