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Question Number 72526 by liki last updated on 30/Oct/19

$$ifthelcmandgcfofthreenumbers$$$$are360and6,othernumbersare18$$$$and60.Findthethirdnumber.$$$$...Ineedhelpplz...$$

Commented by TawaTawa last updated on 30/Oct/19

$$\mathrm{For}\mathrm{three}\mathrm{numbers}$$$$\mathrm{a}\times \mathrm{b}\times \mathrm{c}=\frac{\mathrm{lcm}(\mathrm{a},\mathrm{b},\mathrm{c})\times \gcd (\mathrm{a},\mathrm{b})\times \gcd (\mathrm{c},\mathrm{a})}{\gcd (\mathrm{a},\mathrm{b},\mathrm{c})}$$

Answered by mind is power last updated on 30/Oct/19

$$\gcd (\mathrm{a},\mathrm{b},\mathrm{c})=\mathrm{d}$$$$\mathrm{d}\mid \mathrm{a},\mathrm{b}$$$$\mathrm{d}\mid \mathrm{c}$$$$\mathrm{d}\Rightarrow \mid \gcd (\mathrm{a},\mathrm{b}),\mathrm{c}\Rightarrow \mathrm{d}\mid \gcd ((\gcd (\mathrm{a},\mathrm{b}),\mathrm{c})$$$$\gcd ((\gcd (\mathrm{a},\mathrm{b}),\mathrm{c})=\mathrm{d}\Rightarrow \mathrm{d}\mid \mathrm{c}$$$$\mathrm{d}\mid \gcd (\mathrm{a},\mathrm{b})\Rightarrow \mathrm{d}\mid \mathrm{a}\mathrm{\&}\mathrm{d}\mid \mathrm{b}$$$$\Rightarrow \gcd (\mathrm{a},\mathrm{b},\mathrm{c})=\gcd (\gcd (\mathrm{a},\mathrm{b}),\mathrm{c})$$$$\mathrm{lcm}(\mathrm{a},\mathrm{b},\mathrm{c})=\mathrm{lcm}(\mathrm{lcm}(\mathrm{a},\mathrm{b}),\mathrm{c}))$$$$\mathrm{m}=\mathrm{lcm}(\mathrm{a},\mathrm{b},\mathrm{c})\Rightarrow \mathrm{a}\mid \mathrm{m},\mathrm{b}\mid \mathrm{m}\Rightarrow \mathrm{lcm}(\mathrm{a},\mathrm{b})\mid \mathrm{m}$$$$\mathrm{c}\mid \mathrm{m}\Rightarrow \mathrm{lcm}(\mathrm{lcm}(\mathrm{a},\mathrm{b}),\mathrm{c})\mid \mathrm{m}$$$$\mathrm{d}=\mathrm{lcm}(\mathrm{lcm}(\mathrm{a},\mathrm{b}),\mathrm{c})\Rightarrow \mathrm{c}\mid \mathrm{d},\mathrm{lcm}(\mathrm{a},\mathrm{b})\mid \mathrm{d}\Rightarrow \mathrm{a}\mid \mathrm{d},\mathrm{b}\mid \mathrm{d},\mathrm{c}\mid \mathrm{d}\Rightarrow \mathrm{lcm}(\mathrm{a},\mathrm{b},\mathrm{c})\mid \mathrm{d}$$$$\iff \mathrm{lcm}(\mathrm{lcm}(\mathrm{a},\mathrm{b}),\mathrm{c})=\mathrm{lcm}(\mathrm{a},\mathrm{b},\mathrm{c})$$$$\mathrm{after}\mathrm{that}\mathrm{a},\mathrm{b},\mathrm{c}\mathrm{three}\mathrm{number}$$$$\mathrm{lcm}(\mathrm{a},\mathrm{b},\mathrm{c})=360$$$$\gcd (\mathrm{a},\mathrm{b},\mathrm{c})=18$$$$\mathrm{a}=6,\mathrm{b}=18$$$$\iff \{\begin{array}{l}\mathrm{lcm}(6,18,\mathrm{c})=360=\mathrm{lcm}(\mathrm{lcm}(18,60),\mathrm{c})=360\\ \gcd (60,18,\mathrm{c})=6=\gcd (\gcd (60,18),\mathrm{c})=6\end{array}$$$$\mathrm{lcm}(60,18)=6\mathrm{lcm}(10,3)=180,\gcd (60,18)=6$$$$\Rightarrow \mathrm{lcm}(180,\mathrm{c})=360$$$$\Rightarrow \gcd (6,\mathrm{c})=6$$$$\mathrm{c}=6\mathrm{k}\Rightarrow \gcd (6,6\mathrm{k})=6=6\gcd (\mathrm{k},1)=6$$$$\mathrm{lcm}(180,6\mathrm{k})=360$$$$6\mathrm{lcm}(30,\mathrm{k})=360$$$$\mathrm{lcm}(30,\mathrm{k})=60=\frac{30.\mathrm{k}}{\gcd (30,\mathrm{k})}\Rightarrow 2\gcd (30,\mathrm{k})=\mathrm{k}$$$$\gcd (30,\mathrm{k})\in \{1,2,3,5,6,10,15,30\}$$$$=1\Rightarrow \mathrm{k}=2\mathrm{no}$$$$\gcd =2\Rightarrow \mathrm{k}=4\Rightarrow$$$$\gcd =3\Rightarrow \mathrm{k}=6\mathrm{but}\gcd (6,30)=6\mathrm{no}$$$$\gcd =5\Rightarrow \mathrm{k}=10\gcd (10,30)=10\mathrm{no}$$$$\gcd =6\Rightarrow \mathrm{k}=12$$$$\gcd =10\Rightarrow \mathrm{k}=20$$$$\gcd =15\Rightarrow \mathrm{k}=30\mathrm{non}$$$$\gcd =30\Rightarrow \mathrm{k}=60$$$$\mathrm{k}\in \{4,20,60,12\}$$$$\mathrm{c}=6\mathrm{k}$$$$\mathrm{c}\in \{360,120,24,72\}$$$$$$$$$$

Commented by liki last updated on 30/Oct/19

$$thankssir..$$

Commented by mind is power last updated on 30/Oct/19

$${\mathrm{y}}^{\prime }\mathrm{re}\mathrm{welcom}$$

Commented by mr W last updated on 30/Oct/19

$$24and72alsook$$

Commented by mind is power last updated on 30/Oct/19

$$\mathrm{yes}\mathrm{i}\mathrm{fixed}\mathrm{i}\mathrm{took}6\mathrm{but}[\mathrm{the}\mathrm{two}\mathrm{number}\mathrm{are}18,60\mathrm{not}18\mathrm{and}6$$$$\mathrm{lol}$$

Answered by mr W last updated on 30/Oct/19

$$LCM=360=2^3\times 3^2\times 5^1$$$$GCD=6=2^1\times 3^1$$$$A=18=2^1\times 3^2$$$$B=60=2^2\times 3^1\times 5^1$$$$C=2^3\times 3^{1...2}\times 5^{0...1}$$$$=8\times 3=24$$$$=8\times 3\times 5=120$$$$=8\times 9=72$$$$=8\times 9\times 5=360$$

Commented by TawaTawa last updated on 30/Oct/19

$$\mathrm{Sir}\mathrm{help}\mathrm{me}\mathrm{check}\mathrm{the}\mathrm{question}\mathrm{i}\mathrm{solved}\mathrm{in}\mathrm{Q}72560$$$$\mathrm{if}\mathrm{i}\mathrm{am}\mathrm{correct}$$

Commented by liki last updated on 31/Oct/19

$$\bullet Thankyousir$$

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