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Question Number 191846 by malwan last updated on 01/May/23

find the last three digits of 4^2^(42) Mohammed Alwan

$${find}\:{the}\:{last}\:{three}\:{digits} \\ $$$${of}\:\mathrm{4}^{\mathrm{2}^{\mathrm{42}} } \\ $$$${Mohammed}\:{Alwan} \\ $$

Answered by deleteduser1 last updated on 01/May/23

4^2^(42) =2^2^(43) ≡x(mod 1000)⇒2^(2^(43) −3) ≡(x/8)(mod 125) φ(125)=100 2^(43) −3≡y(mod 100)⇒2^(41) ≡((y+3)/4)(mod 25) φ(25)=20⇒2^(41) ≡(2^(20) )^2 ×2≡1×2≡((y+3)/4)(mod 25) ⇒8≡y+3(mod 100)⇒y≡5(mod 100) ⇒2^(43) ≡8(mod 100) ⇒2^(2^(43) −3) ≡(2^(100q) )2^5 ≡(x/8)(mod 125) ⇒8×32≡x(mod 1000)⇒x≡256(mod 1000) ⇒Last three digits of 4^2^(42) =256

$$\mathrm{4}^{\mathrm{2}^{\mathrm{42}} } =\mathrm{2}^{\mathrm{2}^{\mathrm{43}} } \equiv{x}\left({mod}\:\mathrm{1000}\right)\Rightarrow\mathrm{2}^{\mathrm{2}^{\mathrm{43}} −\mathrm{3}} \equiv\frac{{x}}{\mathrm{8}}\left({mod}\:\mathrm{125}\right) \\ $$$$\phi\left(\mathrm{125}\right)=\mathrm{100} \\ $$$$\mathrm{2}^{\mathrm{43}} −\mathrm{3}\equiv{y}\left({mod}\:\mathrm{100}\right)\Rightarrow\mathrm{2}^{\mathrm{41}} \equiv\frac{{y}+\mathrm{3}}{\mathrm{4}}\left({mod}\:\mathrm{25}\right) \\ $$$$\phi\left(\mathrm{25}\right)=\mathrm{20}\Rightarrow\mathrm{2}^{\mathrm{41}} \equiv\left(\mathrm{2}^{\mathrm{20}} \right)^{\mathrm{2}} ×\mathrm{2}\equiv\mathrm{1}×\mathrm{2}\equiv\frac{{y}+\mathrm{3}}{\mathrm{4}}\left({mod}\:\mathrm{25}\right) \\ $$$$\Rightarrow\mathrm{8}\equiv{y}+\mathrm{3}\left({mod}\:\mathrm{100}\right)\Rightarrow{y}\equiv\mathrm{5}\left({mod}\:\mathrm{100}\right) \\ $$$$\Rightarrow\mathrm{2}^{\mathrm{43}} \equiv\mathrm{8}\left({mod}\:\mathrm{100}\right) \\ $$$$\Rightarrow\mathrm{2}^{\mathrm{2}^{\mathrm{43}} −\mathrm{3}} \equiv\left(\mathrm{2}^{\mathrm{100}{q}} \right)\mathrm{2}^{\mathrm{5}} \equiv\frac{{x}}{\mathrm{8}}\left({mod}\:\mathrm{125}\right) \\ $$$$\Rightarrow\mathrm{8}×\mathrm{32}\equiv{x}\left({mod}\:\mathrm{1000}\right)\Rightarrow{x}\equiv\mathrm{256}\left({mod}\:\mathrm{1000}\right) \\ $$$$\Rightarrow{Last}\:{three}\:{digits}\:{of}\:\mathrm{4}^{\mathrm{2}^{\mathrm{42}} } =\mathrm{256} \\ $$

Commented by malwan last updated on 01/May/23

thank you so much sir

$${thank}\:{you}\:{so}\:{much}\:{sir} \\ $$

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