Question Number 70370 by Rasheed.Sindhi last updated on 04/Oct/19
$${If}\:\:{gcd}\left({p}\:,\:{q}\right)=\mathrm{1},{prove}\:{that} \\ $$$$\:\:\:\:\:{gcd}\left({p}\left({p}+{q}\right)\:,\:{q}\left({p}+{q}\right)\:,\:{pq}\right)=\mathrm{1} \\ $$$$\mathrm{R}\boldsymbol{\mathrm{elated}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{Q}}#\mathrm{69939} \\ $$
Commented by mind is power last updated on 03/Oct/19
$${suppose}\:{That}\:{a}=\:{gcd}\left({p}\left({p}+{q}\right),{q}\left({p}+{q}\right),{pq}\right)\neq\mathrm{1} \\ $$$$\Rightarrow\:\exists\:{prime}\:{number}\:{x}\:{that}\:{x}\mid{a} \\ $$$$\Rightarrow{x}\mid{pq},{x}\mid{q}\left({p}+{q}\right),{x}\mid{p}\left({p}+{q}\right) \\ $$$${x}\mid{pq}+{q}^{\mathrm{2}} ,{x}\mid{pq}+{p}^{\mathrm{2}} ,{x}\mid{pq} \\ $$$$\Rightarrow{x}\mid{q}^{\mathrm{2}} \Rightarrow{x}\mid{q}\:\:{x}\:{is}\:{prime} \\ $$$${x}\mid{p}^{\mathrm{2}} \Rightarrow{x}\mid{p} \\ $$$$\left({x}\mid{p}\:\wedge\:{x}\mid{q}\right)\Leftrightarrow{x}\mid{gcd}\left({p},{q}\right)=\mathrm{1}\Rightarrow{x}\mid\mathrm{1}\:{absurd}\:\:{x}\:{is}\:{prime} \\ $$$$\Rightarrow{ther}\:{exist}\:{not}\:{prime}\:{factor}\:{of}\:{a}\in\mathbb{N}\Leftrightarrow\left({a}=\mathrm{1}\right) \\ $$
Commented by MJS last updated on 03/Oct/19
$$\mathrm{good},\:\mathrm{but}\:\mathrm{I}\:\mathrm{think}\:\mathrm{the}\:\mathrm{last}\:\mathrm{line}\:\mathrm{should}\:\mathrm{be} \\ $$$${x}\mid{p}\:\wedge\:{x}\mid{q}\:\wedge\:\mathrm{gcd}\:\left({p},\:{q}\right)=\mathrm{1}\:\Rightarrow\:{x}=\mathrm{1} \\ $$$$\mathrm{because}\:\mathrm{obviously} \\ $$$$\mathrm{gcd}\:\left({p},\:{q}\right)={r}\:\wedge\:{x}\mid{p}\:\wedge\:{x}\mid{q}\:\Rightarrow\:{x}\leqslant{r} \\ $$
Commented by mind is power last updated on 03/Oct/19
$${x}\mid{a}\wedge{x}\mid{b}\Leftrightarrow{x}\mid{gcd}\left({a},{b}\right) \\ $$$${ifx}\mid{gcd}\left({a},{b}\right)\Rightarrow{x}\mid{a}\wedge{x}\mid{b} \\ $$$${ifx}\mid{a}\wedge{x}\mid{b}\Rightarrow{x}\mid{gcd}\left({a},{b}\right) \\ $$$$ \\ $$
Commented by Rasheed.Sindhi last updated on 04/Oct/19
$$\mathbb{S}\mathrm{ir},\:\mathbb{T}\mathrm{hanks}\:\&\:\mathbb{A}\mathrm{ppreciation}\rightarrow\mathrm{you}! \\ $$
Answered by $@ty@m123 last updated on 04/Oct/19
$${If}\:{possible}\:{let}\:{us}\:{assume}\:{that} \\ $$$$\:{gcd}\left\{{p}\left({p}+{q}\right),{q}\left({p}+{q}\right),{pq}\right\}={a}\neq\mathrm{1} \\ $$$${Then}\:\exists\:{m},{n},{r}\in\mathbb{N}\:{with}\:{gcd}\left({m},{n},{r}\right)=\mathrm{1} \\ $$$${such}\:{that} \\ $$$${p}\left({p}+{q}\right)={a}.{m}\:\_\_\_\left(\mathrm{1}\right) \\ $$$${q}\left({p}+{q}\right)={a}.{n}\:\:\_\_\_\left(\mathrm{2}\right) \\ $$$${pq}={a}.{r}\:\:\_\_\_\left(\mathrm{3}\right) \\ $$$${From}\:\left(\mathrm{1}\right)\:\&\left(\mathrm{3}\right) \\ $$$${p}^{\mathrm{2}} +{ar}={am} \\ $$$${p}^{\mathrm{2}} ={a}\left({m}−{r}\right)\:\_\_\_\left(\mathrm{4}\right) \\ $$$${From}\:\left(\mathrm{2}\right)\:\&\left(\mathrm{3}\right) \\ $$$${ar}+{q}^{\mathrm{2}} ={an} \\ $$$${q}^{\mathrm{2}} ={a}\left({n}−{r}\right)\:\_\_\_\left(\mathrm{5}\right) \\ $$$$\left(\mathrm{4}\right)\:\&\left(\mathrm{5}\right)\Rightarrow{p}^{\mathrm{2}} \mid{a}\:\wedge{q}^{\mathrm{2}} \mid{a} \\ $$$$\Rightarrow{p}\mid{a}\:\wedge{q}\mid{a} \\ $$$$\Rightarrow{gcd}\left({p},{q}\right)={a}\neq\mathrm{1} \\ $$$${which}\:{is}\:{a}\:{contradiction}. \\ $$$$\therefore\:{our}\:{assumption}\:{is}\:{wrong}. \\ $$$$\therefore\:{gcd}\left\{{p}\left({p}+{q}\right),{q}\left({p}+{q}\right),{pq}\right\}=\mathrm{1} \\ $$
Commented by Rasheed.Sindhi last updated on 04/Oct/19
$${Thank}\:{you}\:{sir}.{I}\:{appriciate}\:{your}\:{approach}! \\ $$