Question Number 80580 by mr W last updated on 04/Feb/20
$${Find}\:{general}\:{solution}\:{for}\:{k}\:{such}\:{that} \\ $$$$\mathrm{7}^{{k}} \equiv\mathrm{1}\:{mod}\:\left(\mathrm{35}\right) \\ $$
Answered by Rio Michael last updated on 04/Feb/20
$$\:{k}\:=\:\mathrm{8}{n}\:,\:{n}\:\in\:\mathbb{N} \\ $$
Commented by mr W last updated on 04/Feb/20
$${but}\:\mathrm{7}^{\mathrm{8}{n}} \equiv\mathrm{21}\:{mod}\:\left(\mathrm{35}\right) \\ $$
Commented by mr W last updated on 04/Feb/20
$${i}\:{think}\:{there}\:{is}\:{no}\:{solution}\:{for} \\ $$$$\mathrm{7}^{{k}} \equiv\mathrm{1}\:{mod}\:\left(\mathrm{35}\right) \\ $$$$\mathrm{7}^{{k}} \equiv\mathrm{1}\:{mod}\:\left(\mathrm{55}\right) \\ $$$$\mathrm{7}^{{k}} \equiv\mathrm{1}\:{mod}\:\left(\mathrm{65}\right) \\ $$$$\mathrm{7}^{{k}} \equiv\mathrm{1}\:{mod}\:\left(\mathrm{85}\right) \\ $$$$\mathrm{7}^{{k}} \equiv\mathrm{1}\:{mod}\:\left(\mathrm{95}\right) \\ $$
Commented by Rio Michael last updated on 04/Feb/20
$${ahh}\:{right}\:{sir},{sir}\:{please}\: \\ $$$${can}\:{i}\:{get}\:{your}\:{contact}, \\ $$$${i}\:{would}\:{love}\:{you}\:{to}\:{a}\:{member} \\ $$$${of}\:{a}\:{book}\:{i}\:{will}\:{be}\:{writing}\: \\ $$$$ \\ $$
Commented by mr W last updated on 04/Feb/20
$${thank}\:{you}\:{for}\:{your}\:{kindness}\:{sir}!\: \\ $$$${at}\:{the}\:{moment}\:{i}\:{want}\:{to}\:{restrict}\:{the} \\ $$$${exchange}\:{within}\:{this}\:{forum}.\:{please} \\ $$$${inform}\:{us}\:{when}\:{your}\:{book}\:{is}\:{published}. \\ $$
Commented by Rio Michael last updated on 04/Feb/20
$$\mathrm{but}\:\mathrm{i}\:\mathrm{will}\:\mathrm{really}\:\mathrm{love}\:\mathrm{you}\:\mathrm{to}\:\mathrm{be}\:\mathrm{a}\: \\ $$$$\mathrm{member},\mathrm{like}\:\mathrm{check}\:\mathrm{the}\:\mathrm{solvings} \\ $$$$\mathrm{and}\:\mathrm{do}\:\mathrm{valuable}\:\mathrm{contributions} \\ $$$$\mathrm{you}\:\mathrm{are}\:\mathrm{good}\:\mathrm{sir},\mathrm{very}\:\mathrm{good} \\ $$$$\mathrm{in}\:\mathrm{maths}\:\mathrm{and}\:\mathrm{physics} \\ $$
Commented by otchereabdullai@gmail.com last updated on 05/Feb/20
$$\mathrm{Prof}\:\mathrm{W}\:\mathrm{is}\:\mathrm{the}\:\mathrm{world}\:\mathrm{best}\:\mathrm{mathematician} \\ $$
Answered by rkdbdb last updated on 04/Feb/20
$${k}=\mathrm{0} \\ $$
Commented by mr W last updated on 05/Feb/20
$${that}'{s}\:{true}.\:{but}\:{we}\:{should}\:{find}\:{k}\in{N}. \\ $$