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Question Number 106222 by Rasheed.Sindhi last updated on 03/Aug/20

$$\mathcal{D}eterminex\mathrm{\&}ysuchthat:$$ $$\mathrm{lcm}(x,y)-\gcd (x,y)=x+y.$$

Commented byRasheed.Sindhi last updated on 03/Aug/20

Thanks sir 〽️�� ��

Commented byRasheed.Sindhi last updated on 03/Aug/20

$$Repostedforanswer.$$

Commented bymr W last updated on 03/Aug/20

$$nicequestion!$$ $$$$ $$onekindofsolutions:$$ $$xandyarecoprime,$$ $$gcd(x,y)=1$$ $$lcm(x,y)=xy$$ $$xy-1=x+y$$ $$\Rightarrow y=\frac{x+1}{x-1}=1+\frac{2}{x-1}$$ $$twosolutions:$$ $$x=2,y=3$$ $$x=3,y=2$$ $$$$ $$anotherkindofsolutions:$$ $$x=prime$$ $$y=kx>x$$ $$gcd(x,y)=k$$ $$lcm(x,y)=kx$$ $$kx-k=x+kx$$ $$k=-x<0\Rightarrow nosolution$$

Commented byRasheed.Sindhi last updated on 03/Aug/20

$$ActuallysirIseetheproblemas$$ $$\mathrm{♮}If$$ $$\{\mathrm{lcm}(\mathrm{x},\mathrm{y})-\gcd (\mathrm{x},\mathrm{y})\}:(\mathrm{x}+\mathrm{y})=1:1$$ $$\mathrm{then}\mathrm{x}:\mathrm{y}=2:3\mathrm{or}3:2\epsilon$$ $$\mathrm{This}\mathrm{can}\mathrm{be}\mathrm{generalized}\mathrm{as}:$$ $$\mathrm{♮}\mathrm{If}\{\mathrm{lcm}(\mathrm{x},\mathrm{y})-\gcd (\mathrm{x},\mathrm{y})\}:(\mathrm{x}+\mathrm{y})=\mathrm{p}:\mathrm{q}$$ $$\mathrm{then}\mathrm{m}\mathrm{\&}\mathrm{n}\mathrm{can}\mathrm{be}\mathrm{determined}$$ $$\mathrm{such}\mathrm{that}\mathrm{x}:\mathrm{y}=\mathrm{m}:\mathrm{n}\mathrm{or}\mathrm{n}:\mathrm{m}\epsilon$$ $$$$

Commented bymr W last updated on 03/Aug/20

$$yessir,greatsolution!$$

Answered by Rasheed.Sindhi last updated on 03/Aug/20

$$Let\gcd (x,y)=kand$$ $$x=pk,y=qkwith\gcd (p,q)=1$$ $$i-ep,qarecoprime.$$ $$\mathrm{lcm}(x,y)-\gcd (x,y)=x+y$$ $$\mathrm{lcm}(pk,qk)-\gcd (pk,qk)=pk+qk$$ $$\Rightarrow pqk-k=pk+qk$$ $$\Rightarrow pq-1=p+q$$ $$\Rightarrow p=2,q=3\vee p=3,q=2$$ $$\Rightarrow x=2k,y=3k\vee x=3k,y=2k$$ $$\{x,y\}=\{2k,3k\}$$

Commented byRasheed.Sindhi last updated on 03/Aug/20

$$Allnumberswhichbearratio$$ $$2:3(or3:2)satisfythegiven$$ $$condition.$$

Commented byRasheed.Sindhi last updated on 03/Aug/20

$$\mathcal{S}ir,nosuchcondition.\mathrm{\forall }k\in \mathbb{N}$$ $$SeealsomycommenttomrW$$ $$sir.$$

Commented byRasheed.Sindhi last updated on 03/Aug/20

$$\mathcal{S}irmalwan.Notfork=0,because$$ $$itdestroysratio(2:3or3:2).In$$ $$additionthegcd(0,0)isnot$$ $$defined.$$ $$SeealsomycommenttomrW$$ $$sir.$$

Commented byPRITHWISH SEN 2 last updated on 03/Aug/20

$$\mathrm{Nice}\mathrm{sir}$$

Commented bymalwaan last updated on 03/Aug/20

$$Isittruefork=0?$$

Commented byPRITHWISH SEN 2 last updated on 03/Aug/20

$$\mathrm{Is}\mathrm{there}\mathrm{any}\mathrm{certain}\mathrm{condition}\mathrm{that}\mathrm{k}\mathrm{should}\mathrm{be}$$ $$\mathrm{prime}\mathrm{or}\mathrm{it}\mathrm{is}\mathrm{true}\mathrm{for}\mathrm{all}\mathrm{natural}\mathrm{numbers}?$$

Commented bymalwaan last updated on 04/Aug/20

$$thankyousomuchsir$$

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