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Question Number 205842 by BaliramKumar last updated on 31/Mar/24

$$\frac{{32}^{{32}^{32}}}{9}\stackrel{\mathrm{R}}{\equiv }?$$

Answered by A5T last updated on 01/Apr/24

$${32}^{{32}^{32}}\stackrel{9}{\equiv }{5}^{{32}^{32}};5^3\stackrel{9}{\equiv }-1\Rightarrow {32}^{{32}^{32}}=5^{3k+1}\equiv {\left(5^3\right)}^k\left(5\right)$$$$\equiv {(-1)}^k\left(5\right)\equiv -5\left(sincekisodd\right)\stackrel{9}{\equiv }4$$

Answered by A5T last updated on 01/Apr/24

$${32}^{{32}^{32}}\stackrel{9}{\equiv }{4}^{{32}^{32}}$$$$4^3\equiv 64\stackrel{9}{\equiv }1\Rightarrow 4^{{32}^{32}}\equiv 4^{{(-1)}^{32}}\stackrel{9}{\equiv }4$$

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