Question Number 23968 by Tinkutara last updated on 10/Nov/17
$${Prove}\:{that}\:{n}\:{and}\:\mathrm{2}{n}−\mathrm{1}\:{are}\:{coprime}. \\ $$
Commented by ajfour last updated on 10/Nov/17
$${thanks}. \\ $$
Commented by prakash jain last updated on 10/Nov/17
$${l}={GCD}\left({n},\mathrm{2}{n}−\mathrm{1}\right) \\ $$$${n}={kl},\mathrm{2}{n}−\mathrm{1}={ml}\: \\ $$$$\mathrm{2}{n}−\mathrm{1}=\mathrm{2}{kl}−\mathrm{1}={ml} \\ $$$$\Rightarrow\left(\mathrm{2}{k}−{m}\right){l}=\mathrm{1}\Rightarrow\mathrm{2}{k}−{m}=\mathrm{1},{l}=\mathrm{1} \\ $$$${l}=\mathrm{1} \\ $$
Commented by Tinkutara last updated on 10/Nov/17
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$