Find-out-x-y-such-that-lcm-x-y-180-gcd-x-y-45-

Question Number 62242 by Rasheed.Sindhi last updated on 18/Jun/19

$$\mathrm{Find}\:\mathrm{out}\:\mathrm{x},\mathrm{y}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{lcm}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{180}\:\wedge\:\mathrm{gcd}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{45} \\ $$

Commented by maxmathsup by imad last updated on 19/Jun/19

Δ(x,y) =45 and M(x,y)=180 ⇒x =45 u and y =45 v with Δ(u,v)=1 M(x,y) =180 ⇒M(45u,45v)=180 ⇒45M(u,v) =180 ⇒M(u,v) =4 u.v =M(u,v).Δ(u,v) =4 ⇒xy =45^2 .4⇒xy =4×45×45 ....

$$\Delta\left({x},{y}\right)\:=\mathrm{45}\:{and}\:{M}\left({x},{y}\right)=\mathrm{180}\:\Rightarrow{x}\:=\mathrm{45}\:{u}\:\:{and}\:{y}\:=\mathrm{45}\:{v}\:{with}\:\Delta\left({u},{v}\right)=\mathrm{1} \\ $$$${M}\left({x},{y}\right)\:=\mathrm{180}\:\Rightarrow{M}\left(\mathrm{45}{u},\mathrm{45}{v}\right)=\mathrm{180}\:\Rightarrow\mathrm{45}{M}\left({u},{v}\right)\:=\mathrm{180}\:\Rightarrow{M}\left({u},{v}\right)\:=\mathrm{4} \\ $$$${u}.{v}\:={M}\left({u},{v}\right).\Delta\left({u},{v}\right)\:=\mathrm{4}\:\Rightarrow{xy}\:=\mathrm{45}^{\mathrm{2}} .\mathrm{4}\Rightarrow{xy}\:=\mathrm{4}×\mathrm{45}×\mathrm{45}\:…. \\ $$

Answered by malwaan last updated on 18/Jun/19

always (gcd ; lcm) is a solution so (x;y)=(45;180)

$${always}\:\left({gcd}\:;\:{lcm}\right)\:{is}\:{a}\:{solution} \\ $$$${so}\:\left({x};{y}\right)=\left(\mathrm{45};\mathrm{180}\right) \\ $$

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

$${Appreciation}\:{for}\:{rapid}\:{response}!\: \\ $$$${But}\:{there}\:{may}\:{be}\:{other}\:{solutions}\:{too}.{Try} \\ $$$${to}\:{find}\:{out}. \\ $$

Commented by malwaan last updated on 18/Jun/19

for this question I am sure that is the only solution but If d=12 and m=432 we will get tow solutions {{12 ; 432} ; {48 ; 108}}

$${for}\:{this}\:{question}\:\: \\ $$$${I}\:{am}\:{sure}\:{that}\:{is}\:{the}\:{only}\:{solution} \\ $$$${but}\:{If}\:\:{d}=\mathrm{12}\:{and}\:{m}=\mathrm{432} \\ $$$${we}\:{will}\:{get}\:{tow}\:{solutions} \\ $$$$\left\{\left\{\mathrm{12}\:;\:\mathrm{432}\right\}\:;\:\left\{\mathrm{48}\:;\:\mathrm{108}\right\}\right\} \\ $$

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

You′re right sir! I′m sorry for my wrong viewpoint.Actually I made my question for multi-solutions but after that I changed my numbers ignorantly!

$$\mathrm{You}'\mathrm{re}\:\mathrm{right}\:\mathrm{sir}!\:\mathrm{I}'\mathrm{m}\:\mathrm{sorry}\:\mathrm{for}\:\mathrm{my}\:\mathrm{wrong} \\ $$$$\mathrm{viewpoint}.\mathrm{Actually}\:\mathrm{I}\:\mathrm{made}\:\mathrm{my}\:\mathrm{question} \\ $$$$\mathrm{for}\:\mathrm{multi}-\mathrm{solutions}\:\mathrm{but}\:\mathrm{after}\:\mathrm{that}\:\mathrm{I}\:\mathrm{changed} \\ $$$$\mathrm{my}\:\mathrm{numbers}\:\mathrm{ignorantly}! \\ $$

Answered by malwaan last updated on 18/Jun/19

another way x= 45a ; y= 45b ⇒45ab=180⇒ab= 4 ⇒a=1;b=4 (coprime) ⇒x=45×1= 45 ; y=45×4 = 180 ∴ (x;y)=(45;180)

$${another}\:{way} \\ $$$${x}=\:\mathrm{45}{a}\:;\:{y}=\:\mathrm{45}{b} \\ $$$$\Rightarrow\mathrm{45}{ab}=\mathrm{180}\Rightarrow{ab}=\:\mathrm{4} \\ $$$$\Rightarrow{a}=\mathrm{1};{b}=\mathrm{4}\:\left({coprime}\right) \\ $$$$\Rightarrow{x}=\mathrm{45}×\mathrm{1}=\:\mathrm{45} \\ $$$$\:;\:{y}=\mathrm{45}×\mathrm{4}\:=\:\mathrm{180} \\ $$$$\therefore\:\left({x};{y}\right)=\left(\mathrm{45};\mathrm{180}\right) \\ $$

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

$$\mathcal{T}{hank}\:{you}\:\mathcal{S}{ir}! \\ $$

Commented by malwaan last updated on 19/Jun/19

$${most}\:{welcome}\: \\ $$

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