Question Number 106017 by Rasheed.Sindhi last updated on 02/Aug/20
$$\mathcal{D}{etermine}\:{x}\:\&\:{y},\:{such}\:{that}: \\ $$$$\mathrm{lcm}\left({x},{y}\right)−\mathrm{gcd}\left({x},{y}\right)={x}+{y}. \\ $$
Answered by Rasheed.Sindhi last updated on 03/Aug/20
$${Let}\:\mathrm{gcd}\left({x},{y}\right)={k}\:{and} \\ $$$$\:\:\:\:{x}={pk}\:,\:{y}={qk}\:\:{with}\:\mathrm{gcd}\left({p},{q}\right)=\mathrm{1} \\ $$$${i}-{e}\:{p},{q}\:{are}\:{coprime}. \\ $$$$\:\mathrm{lcm}\left({x},{y}\right)−\mathrm{gcd}\left({x},{y}\right)={x}+{y} \\ $$$$\:\mathrm{lcm}\left({pk},{qk}\right)−\mathrm{gcd}\left({pk},{qk}\right)={pk}+{qk} \\ $$$$\:\:\:\:\:\Rightarrow{pqk}−{k}={pk}+{qk} \\ $$$$\:\:\:\:\:\Rightarrow{pq}−\mathrm{1}={p}+{q} \\ $$$$\:\:\Rightarrow{p}=\mathrm{2}\:,\:{q}=\mathrm{3}\:\:\:\vee\:\:{p}=\mathrm{3}\:,\:{q}=\mathrm{2} \\ $$$$\:\:\Rightarrow{x}=\mathrm{2}{k}\:,{y}=\mathrm{3}{k}\:\:\vee\:{x}=\mathrm{3}{k},{y}=\mathrm{2}{k} \\ $$$$\left\{{x},{y}\right\}=\left\{\mathrm{2}{k},\mathrm{3}{k}\right\} \\ $$