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Question Number 33089 by mondodotto@gmail.com last updated on 10/Apr/18

The LCM and GCF of three numbers is 360 and 6 respectively. if the two numbers are 18 and 60. find the third number.

$$\:\boldsymbol{\mathrm{T}}\mathrm{he}\:\boldsymbol{\mathrm{LCM}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{GCF}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{three}}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{is}} \\ $$$$\:\mathrm{360}\:\boldsymbol{\mathrm{and}}\:\mathrm{6}\:\boldsymbol{\mathrm{respectively}}.\:\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{the}} \\ $$$$\:\boldsymbol{\mathrm{two}}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{are}}\:\mathrm{18}\:\boldsymbol{\mathrm{and}}\:\mathrm{60}. \\ $$$$\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{third}}\:\boldsymbol{\mathrm{number}}. \\ $$

Commented by Rasheed.Sindhi last updated on 10/Apr/18

24,72,120,360

$$\mathrm{24},\mathrm{72},\mathrm{120},\mathrm{360} \\ $$

Commented by mondodotto@gmail.com last updated on 10/Apr/18

please show workings

$$\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{workings}} \\ $$

Commented by mondodotto@gmail.com last updated on 10/Apr/18

please help

$$\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{help}} \\ $$

Commented by mondodotto@gmail.com last updated on 10/Apr/18

solution please

$$\mathrm{solution}\:\mathrm{please} \\ $$

Commented by mondodotto@gmail.com last updated on 10/Apr/18

also answers are not true because i have tried to prove i get LCM=720 instead of 360 but GCF is correct

$$\mathrm{also}\:\mathrm{answers}\:\mathrm{are}\:\mathrm{not}\:\mathrm{true}\:\mathrm{because}\:\mathrm{i}\:\mathrm{have}\:\mathrm{tried}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{i}\:\mathrm{get}\:\mathrm{LCM}=\mathrm{720}\:\mathrm{instead}\:\mathrm{of}\:\mathrm{360}\:\mathrm{but}\:\mathrm{GCF}\:\mathrm{is}\:\mathrm{correct} \\ $$

Answered by MJS last updated on 10/Apr/18

18=2×3^2 60=2^2 ×3×5 6=2×3 ⇒ x=2^l ×3^m ; l, m>0 360=2^3 ×3^2 ×5 ⇒ x=2^3 ×3^m ×5^n ; 0≤m≤2; 0≤n≤1 x_1 =2^3 ×3=24 x_2 =2^3 ×3^2 =72 x_3 =2^3 ×3×5=120 x_4 =2^3 ×3^2 ×5=360

$$\mathrm{18}=\mathrm{2}×\mathrm{3}^{\mathrm{2}} \\ $$$$\mathrm{60}=\mathrm{2}^{\mathrm{2}} ×\mathrm{3}×\mathrm{5} \\ $$$$\mathrm{6}=\mathrm{2}×\mathrm{3}\:\Rightarrow\:{x}=\mathrm{2}^{{l}} ×\mathrm{3}^{{m}} ;\:{l},\:{m}>\mathrm{0} \\ $$$$\mathrm{360}=\mathrm{2}^{\mathrm{3}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{5}\:\Rightarrow\:{x}=\mathrm{2}^{\mathrm{3}} ×\mathrm{3}^{{m}} ×\mathrm{5}^{{n}} ;\:\mathrm{0}\leqslant{m}\leqslant\mathrm{2};\:\mathrm{0}\leqslant{n}\leqslant\mathrm{1} \\ $$$${x}_{\mathrm{1}} =\mathrm{2}^{\mathrm{3}} ×\mathrm{3}=\mathrm{24} \\ $$$${x}_{\mathrm{2}} =\mathrm{2}^{\mathrm{3}} ×\mathrm{3}^{\mathrm{2}} =\mathrm{72} \\ $$$${x}_{\mathrm{3}} =\mathrm{2}^{\mathrm{3}} ×\mathrm{3}×\mathrm{5}=\mathrm{120} \\ $$$${x}_{\mathrm{4}} =\mathrm{2}^{\mathrm{3}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{5}=\mathrm{360} \\ $$

Commented by mondodotto@gmail.com last updated on 10/Apr/18

120 and 360 proved correct but 24 and 72 are not

$$\mathrm{120}\:\mathrm{and}\:\mathrm{360}\:\mathrm{proved}\:\mathrm{correct} \\ $$$$\mathrm{but}\:\mathrm{24}\:\mathrm{and}\:\mathrm{72}\:\mathrm{are}\:\mathrm{not} \\ $$

Commented by MJS last updated on 10/Apr/18

then your proof is wrong the gcd(18;60;x); x∈{24;72;120;360} is 6 because the gcd(18;60)=6 and it can′t get higher adding a 3^(rd) number lcm(18;24;60)= =lcm(2×3^2 ;2^3 ×3;2^2 ×3×5)=360 lcm(18;60;72)= =lcm(2×3^2 ;2^2 ×3×5;2^3 ×3^2 )=360 lcm(18;60;120)= =lcm(2×3^2 ;2^2 ×3×5;2^3 ×3×5)=360 lcm(18;60;360)= =lcm(2×3^2 ;2^2 ×3×5;2^3 ×3^2 ×5)=360

$$\mathrm{then}\:\mathrm{your}\:\mathrm{proof}\:\mathrm{is}\:\mathrm{wrong} \\ $$$$\mathrm{the}\:{gcd}\left(\mathrm{18};\mathrm{60};{x}\right);\:{x}\in\left\{\mathrm{24};\mathrm{72};\mathrm{120};\mathrm{360}\right\} \\ $$$$\mathrm{is}\:\mathrm{6}\:\mathrm{because}\:\mathrm{the}\:{gcd}\left(\mathrm{18};\mathrm{60}\right)=\mathrm{6} \\ $$$$\mathrm{and}\:\mathrm{it}\:\mathrm{can}'\mathrm{t}\:\mathrm{get}\:\mathrm{higher}\:\mathrm{adding} \\ $$$$\mathrm{a}\:\mathrm{3}^{\mathrm{rd}} \:\mathrm{number} \\ $$$$ \\ $$$${lcm}\left(\mathrm{18};\mathrm{24};\mathrm{60}\right)= \\ $$$$={lcm}\left(\mathrm{2}×\mathrm{3}^{\mathrm{2}} ;\mathrm{2}^{\mathrm{3}} ×\mathrm{3};\mathrm{2}^{\mathrm{2}} ×\mathrm{3}×\mathrm{5}\right)=\mathrm{360} \\ $$$${lcm}\left(\mathrm{18};\mathrm{60};\mathrm{72}\right)= \\ $$$$={lcm}\left(\mathrm{2}×\mathrm{3}^{\mathrm{2}} ;\mathrm{2}^{\mathrm{2}} ×\mathrm{3}×\mathrm{5};\mathrm{2}^{\mathrm{3}} ×\mathrm{3}^{\mathrm{2}} \right)=\mathrm{360} \\ $$$${lcm}\left(\mathrm{18};\mathrm{60};\mathrm{120}\right)= \\ $$$$={lcm}\left(\mathrm{2}×\mathrm{3}^{\mathrm{2}} ;\mathrm{2}^{\mathrm{2}} ×\mathrm{3}×\mathrm{5};\mathrm{2}^{\mathrm{3}} ×\mathrm{3}×\mathrm{5}\right)=\mathrm{360} \\ $$$${lcm}\left(\mathrm{18};\mathrm{60};\mathrm{360}\right)= \\ $$$$={lcm}\left(\mathrm{2}×\mathrm{3}^{\mathrm{2}} ;\mathrm{2}^{\mathrm{2}} ×\mathrm{3}×\mathrm{5};\mathrm{2}^{\mathrm{3}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{5}\right)=\mathrm{360} \\ $$

Commented by mondodotto@gmail.com last updated on 10/Apr/18

thanx now i understand

$$\mathrm{thanx}\:\mathrm{now}\:\mathrm{i}\:\mathrm{understand} \\ $$

Commented by MJS last updated on 10/Apr/18

you′re welcome

$$\mathrm{you}'\mathrm{re}\:\mathrm{welcome} \\ $$

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