Question Number 144961 by mathdanisur last updated on 30/Jun/21
$${x}\in\mathbb{Z}^{+} \\ $$$$\mathrm{15}^{\mathrm{48}\boldsymbol{{a}}+\mathrm{1}} \:\equiv\:{x}\:\left({mod}\:\mathrm{17}\right)\:\:{find}\:\:{x}=?\: \\ $$
Answered by Rasheed.Sindhi last updated on 01/Jul/21
$$\mathrm{15}^{\mathrm{8}} \equiv{x}\left({mod}\mathrm{17}\right) \\ $$$$\mathrm{15}^{\mathrm{8}} \equiv\mathrm{1}\left({mod}\mathrm{17}\right) \\ $$$$\left(\mathrm{15}^{\mathrm{8}} \right)^{\mathrm{6}{a}} \equiv\left(\mathrm{1}\right)^{\mathrm{6}{a}} \left({mod}\mathrm{17}\right) \\ $$$$\mathrm{15}^{\mathrm{48}{a}} \equiv\mathrm{1} \\ $$$$\mathrm{15}^{\mathrm{48}{a}} .\mathrm{15}\equiv\mathrm{1}.\mathrm{15}\left({mod}\mathrm{17}\right) \\ $$$$\mathrm{15}^{\mathrm{48}{a}+\mathrm{1}} \equiv\mathrm{15}\left({mod}\mathrm{17}\right) \\ $$$${x}\equiv\mathrm{15}\left({mod}\mathrm{17}\right) \\ $$$${x}=\mathrm{15}+\mathrm{17}{k};\:{k}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},....\right\} \\ $$
Commented by mathdanisur last updated on 01/Jul/21
$${cool}\:{Sir}\:{thank}\:{you},\:{answer}\:{x}=\mathrm{15}? \\ $$
Commented by Rasheed.Sindhi last updated on 01/Jul/21
$$\:\mathrm{15}+\mathrm{17}{k};\:{Where}\:{k}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},...\right\} \\ $$
Commented by mathdanisur last updated on 01/Jul/21
$${Sir},\:{but}\:{answer}:\:\mathrm{15} \\ $$
Commented by Rasheed.Sindhi last updated on 01/Jul/21
$$\mathcal{T}{he}\:{least}\:{positive}\:{integral}\:{value} \\ $$$${of}\:\:{x}\:{is}\:\mathrm{15} \\ $$
Commented by mathdanisur last updated on 01/Jul/21
$${thanks}\:{Sir} \\ $$