Question Number 187619 by BaliramKumar last updated on 19/Feb/23
$$\mathrm{32}^{\mathrm{32}^{\mathrm{32}} } \:\equiv\:{r}\:\left({mod}\:\:\:\mathrm{7}\right) \\ $$$${r}\:=\:? \\ $$
Answered by SEKRET last updated on 19/Feb/23
$$\left(\mathrm{32}\right)^{\mathrm{32}^{\mathrm{32}} } \equiv\boldsymbol{\mathrm{r}}\:\boldsymbol{\mathrm{mod}}\left(\mathrm{7}\right)\:\:\:\:\:\:\boldsymbol{\mathrm{r}}=? \\ $$$$\:\:\left(\mathrm{4}\centerdot\mathrm{7}+\mathrm{4}\right)^{\mathrm{32}^{\mathrm{32}\:} } \\ $$$$\:\:\mathrm{4}^{\mathrm{1}} \equiv\mathrm{4}\:\boldsymbol{\mathrm{mod}}\left(\mathrm{7}\right) \\ $$$$\:\mathrm{4}^{\mathrm{2}} \equiv\mathrm{2}\:\boldsymbol{\mathrm{mod}}\left(\mathrm{7}\right) \\ $$$$\:\:\mathrm{4}^{\mathrm{3}} \equiv\mathrm{1}\:\boldsymbol{\mathrm{mod}}\left(\mathrm{7}\right) \\ $$$$\:\:\mathrm{4}^{\mathrm{4}} \equiv\mathrm{4}\:\boldsymbol{\mathrm{mod}}\left(\mathrm{7}\right)\rightarrow\:\:\boldsymbol{\mathrm{T}}=\mathrm{3} \\ $$$$\:\:\:\:\mathrm{32}^{\mathrm{32}} \equiv\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{mod}}\left(\mathrm{3}\right) \\ $$$$\:\:\:\left(\mathrm{3}\centerdot\mathrm{10}+\mathrm{2}\right)^{\mathrm{32}} \\ $$$$\:\:\:\mathrm{2}^{\mathrm{1}} \equiv\mathrm{2}\:\boldsymbol{\mathrm{mod}}\left(\mathrm{3}\right) \\ $$$$\:\:\:\mathrm{2}^{\mathrm{2}} \equiv\mathrm{1}\:\boldsymbol{\mathrm{mod}}\left(\mathrm{3}\right) \\ $$$$\:\:\:\mathrm{2}^{\mathrm{3}} \equiv\:\mathrm{2}\:\boldsymbol{\mathrm{mod}}\left(\mathrm{3}\right)\:\rightarrow\:\:\boldsymbol{\mathrm{T}}_{\mathrm{2}} =\mathrm{2} \\ $$$$\:\:\mathrm{32}\equiv\mathrm{0}\:\boldsymbol{\mathrm{mod}}\left(\mathrm{2}\right) \\ $$$$\:\:\mathrm{32}^{\mathrm{1}} \equiv\:\mathrm{4}\:\boldsymbol{\mathrm{mod}}\left(\mathrm{7}\right) \\ $$$$\:\:\mathrm{32}^{\mathrm{32}^{\mathrm{32}} } =\:\mathrm{4}\:\boldsymbol{\mathrm{mod}}\left(\mathrm{7}\right) \\ $$$$\:\:\boldsymbol{{ABDULAZIZ}}\:\:\boldsymbol{{ABDUVALIYEV}} \\ $$$$ \\ $$
Commented by BaliramKumar last updated on 20/Feb/23
$${Thanks}\:{Sir} \\ $$