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32-32-32-9-R-




Question Number 205842 by BaliramKumar last updated on 31/Mar/24
((32^(32^(32) ) )/9) ≡^R  ?
$$\frac{\mathrm{32}^{\mathrm{32}^{\mathrm{32}} } }{\mathrm{9}}\:\overset{\mathrm{R}} {\equiv}\:? \\ $$
Answered by A5T last updated on 01/Apr/24
32^(32^(32) ) ≡^9 5^(32^(32) ) ; 5^3 ≡^9 −1⇒32^(32^(32) ) =5^(3k+1) ≡(5^3 )^k (5)  ≡(−1)^k (5)≡−5(since k is odd)≡^9 4
$$\mathrm{32}^{\mathrm{32}^{\mathrm{32}} } \overset{\mathrm{9}} {\equiv}\mathrm{5}^{\mathrm{32}^{\mathrm{32}} } ;\:\mathrm{5}^{\mathrm{3}} \overset{\mathrm{9}} {\equiv}−\mathrm{1}\Rightarrow\mathrm{32}^{\mathrm{32}^{\mathrm{32}} } =\mathrm{5}^{\mathrm{3}{k}+\mathrm{1}} \equiv\left(\mathrm{5}^{\mathrm{3}} \right)^{{k}} \left(\mathrm{5}\right) \\ $$$$\equiv\left(−\mathrm{1}\right)^{{k}} \left(\mathrm{5}\right)\equiv−\mathrm{5}\left({since}\:{k}\:{is}\:{odd}\right)\overset{\mathrm{9}} {\equiv}\mathrm{4} \\ $$
Answered by A5T last updated on 01/Apr/24
32^(32^(32) ) ≡^9 4^(32^(32) )   4^3 ≡64≡^9 1⇒4^(32^(32) ) ≡4^((−1)^(32) ) ≡^9 4
$$\mathrm{32}^{\mathrm{32}^{\mathrm{32}} } \overset{\mathrm{9}} {\equiv}\mathrm{4}^{\mathrm{32}^{\mathrm{32}} } \\ $$$$\mathrm{4}^{\mathrm{3}} \equiv\mathrm{64}\overset{\mathrm{9}} {\equiv}\mathrm{1}\Rightarrow\mathrm{4}^{\mathrm{32}^{\mathrm{32}} } \equiv\mathrm{4}^{\left(−\mathrm{1}\right)^{\mathrm{32}} } \overset{\mathrm{9}} {\equiv}\mathrm{4} \\ $$

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