Question Number 159163 by mkam last updated on 13/Nov/21
Commented by mkam last updated on 13/Nov/21
$$?????? \\ $$
Answered by gsk2684 last updated on 14/Nov/21
$${a}\mid{b}\:{means}\:{a}\:{devides}\:{b} \\ $$
Commented by mkam last updated on 14/Nov/21
$${yes}\:{sir} \\ $$
Answered by aleks041103 last updated on 14/Nov/21
$${Q}\mathrm{1}. \\ $$$${a}\mid{b}\:\Rightarrow\:{b}={na},\:{n}\in\mathbb{Z} \\ $$$${b}\mid{a}\Rightarrow{a}={mb},\:{m}\in\mathbb{Z} \\ $$$$\Rightarrow{a}={mna}\Rightarrow{mn}=\mathrm{1} \\ $$$${Since}\:{m},{n}\in\mathbb{Z},\:{then}\: \\ $$$${m}={n}=−\mathrm{1}\:{and}\:{a}=−{b} \\ $$$${or} \\ $$$${m}={n}=\mathrm{1}\:{and}\:{a}={b} \\ $$$$ \\ $$$${Q}\mathrm{2}. \\ $$$$\left.\mathrm{1}\right){a}\mid{b}\Rightarrow{ma}\mid{mb} \\ $$$${a}\mid{b}\Rightarrow{b}={na} \\ $$$$\Rightarrow{mb}={mna}={n}\left({ma}\right)=\left({mb}\right) \\ $$$$\Rightarrow{ma}\mid{mb}. \\ $$$$\left.\mathrm{2}\right){ma}\mid{mb}\Rightarrow{a}\mid{b} \\ $$$${ma}\mid{mb}\Rightarrow{mb}={n}\left({ma}\right) \\ $$$${m}\neq\mathrm{0}\Rightarrow{b}={na}\Rightarrow{a}\mid{b}. \\ $$$$\left.{Summary}\right) \\ $$$${a}\mid{b}\Rightarrow{ma}\mid{mb}\:{and}\:{ma}\mid{mb}\Rightarrow{a}\mid{b} \\ $$$${Then}\:{a}\mid{b}\Leftrightarrow{ma}\mid{mb}. \\ $$