Question Number 73553 by 01 last updated on 13/Nov/19
$$\mathrm{give}:\:\mathrm{a},\mathrm{b},\mathrm{c}>\mathrm{0} \\ $$ $$\mathrm{if}\:\mathrm{a}\mid\mathrm{b}\:\mathrm{and}\:\mathrm{b}\mid\mathrm{c} \\ $$ $$\mathrm{prove}\::\:\mathrm{a}\mid\mathrm{c} \\ $$
Answered by Rio Michael last updated on 13/Nov/19
$${a}\mid{b}\:\Leftrightarrow\:\exists\:{k}\in\mathbb{Z}\::\:\:{b}\:=\:{ka}\:......\left({i}\right) \\ $$ $${b}\mid\:{c}\:\Leftrightarrow\:\exists\:{m}\:\in\mathbb{Z}\::\:\:{c}\:=\:{bm}.....\left({ii}\right) \\ $$ $${eqn}\:\left({i}\right)\:{in}\:\left({ii}\right)\:\Rightarrow \\ $$ $$\:\:{c}\:=\:\left({ka}\right){m} \\ $$ $$\:\:{c}\:=\:{km}\:{a}\:\: \\ $$ $${if}\:{k},{m}\in\:\mathbb{Z}\:{the}\:{km}\in\mathbb{Z} \\ $$ $${hence}\:{a}\mid{c} \\ $$ $$ \\ $$