Question Number 124944 by udaythool last updated on 07/Dec/20
![1. (a, m)=(b, m)=1⇒(ab, m)=1 2. c∣ab and (c, a)=1⇒c∣b 3. If c is a common multiple of a and b then [a, b]∣c 4. [ma, mb]=m[a, b] for all int m>0 5. [a, b](a, b)=∣ab∣ 6. Let g>0, s be integers. Show that g∣s iff ∃ integers x, y such that s=x+y and (x, y)=g](https://www.tinkutara.com/question/Q124944.png)
$$\mathrm{1}.\:\left({a},\:{m}\right)=\left({b},\:{m}\right)=\mathrm{1}\Rightarrow\left({ab},\:{m}\right)=\mathrm{1} \\ $$$$\mathrm{2}.\:{c}\mid{ab}\:\mathrm{and}\:\left({c},\:{a}\right)=\mathrm{1}\Rightarrow{c}\mid{b} \\ $$$$\mathrm{3}.\:\mathrm{If}\:{c}\:\mathrm{is}\:\mathrm{a}\:\mathrm{common}\:\mathrm{multiple}\:\mathrm{of} \\ $$$${a}\:\mathrm{and}\:{b}\:\mathrm{then}\:\left[{a},\:{b}\right]\mid{c} \\ $$$$\mathrm{4}.\:\left[{ma},\:{mb}\right]={m}\left[{a},\:{b}\right]\:\mathrm{for}\:\mathrm{all}\:\mathrm{int}\:{m}>\mathrm{0} \\ $$$$\mathrm{5}.\:\left[{a},\:{b}\right]\left({a},\:{b}\right)=\mid{ab}\mid \\ $$$$\mathrm{6}.\:\mathrm{Let}\:{g}>\mathrm{0},\:{s}\:\mathrm{be}\:\mathrm{integers}.\:\mathrm{Show} \\ $$$$\mathrm{that}\:{g}\mid{s}\:\mathrm{iff}\:\exists\:\mathrm{integers}\:{x},\:{y}\:\mathrm{such} \\ $$$$\mathrm{that}\:{s}={x}+{y}\:\mathrm{and}\:\left({x},\:{y}\right)={g} \\ $$