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Question Number 62214 by Rasheed.Sindhi last updated on 17/Jun/19

Find out x,y such that              ((lcm(x,y))/(gcd(x,y)))=lcm(x,y)−gcd(x,y)

$$\mathrm{Find}\:\mathrm{out}\:\mathrm{x},\mathrm{y}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{lcm}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{gcd}\left(\mathrm{x},\mathrm{y}\right)}=\mathrm{lcm}\left(\mathrm{x},\mathrm{y}\right)−\mathrm{gcd}\left(\mathrm{x},\mathrm{y}\right) \\ $$

Answered by MJS last updated on 17/Jun/19

lcm (x, y)=((gcd^2  (x, y))/(gcd (x,y) −1))  ⇒ gcd (x,y)=2∧lcm (x, y)=4 ⇒   ⇒ x=2∧y=4 ∨ x=4∧y=2

$$\mathrm{lcm}\:\left({x},\:{y}\right)=\frac{\mathrm{gcd}^{\mathrm{2}} \:\left({x},\:{y}\right)}{\mathrm{gcd}\:\left({x},{y}\right)\:−\mathrm{1}} \\ $$$$\Rightarrow\:\mathrm{gcd}\:\left({x},{y}\right)=\mathrm{2}\wedge\mathrm{lcm}\:\left({x},\:{y}\right)=\mathrm{4}\:\Rightarrow\: \\ $$$$\Rightarrow\:{x}=\mathrm{2}\wedge{y}=\mathrm{4}\:\vee\:{x}=\mathrm{4}\wedge{y}=\mathrm{2} \\ $$

Commented by Rasheed.Sindhi last updated on 18/Jun/19

THαnk you Sir!

$$\mathcal{TH}\alpha{nk}\:{you}\:\mathcal{S}{ir}! \\ $$

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