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Question Number 41167 by Cheyboy last updated on 02/Aug/18

The LCM and the GCF of three intergers are 180 and 3 respectively. Two numbers are 45 and 60. What is the third number

$${The}\:{LCM}\:{and}\:{the}\:{GCF}\:{of}\:{three} \\ $$$${intergers}\:{are}\:\mathrm{180}\:{and}\:\mathrm{3}\:{respectively}. \\ $$$${Two}\:{numbers}\:{are}\:\mathrm{45}\:{and}\:\mathrm{60}. \\ $$$${What}\:{is}\:{the}\:{third}\:{number} \\ $$

Answered by MJS last updated on 02/Aug/18

45=3^2 5 60=2^2 3×5 lcm(45, 60)=2^2 3^2 5=180 lcm(180, x)=180 ⇒ x=2^(0≤i≤2) 3^(0≤j≤2) 5^(0≤k≤1) gcf(45, 60)=3×5=15 gcf(15, x)=3 ⇒ x=2^(0≤i) 3^(1≤j) ⇒ x∈{3, 6, 9, 12, 18, 36}

$$\mathrm{45}=\mathrm{3}^{\mathrm{2}} \mathrm{5} \\ $$$$\mathrm{60}=\mathrm{2}^{\mathrm{2}} \mathrm{3}×\mathrm{5} \\ $$$$\mathrm{lcm}\left(\mathrm{45},\:\mathrm{60}\right)=\mathrm{2}^{\mathrm{2}} \mathrm{3}^{\mathrm{2}} \mathrm{5}=\mathrm{180} \\ $$$$\mathrm{lcm}\left(\mathrm{180},\:{x}\right)=\mathrm{180} \\ $$$$\Rightarrow\:{x}=\mathrm{2}^{\mathrm{0}\leqslant{i}\leqslant\mathrm{2}} \mathrm{3}^{\mathrm{0}\leqslant{j}\leqslant\mathrm{2}} \mathrm{5}^{\mathrm{0}\leqslant{k}\leqslant\mathrm{1}} \\ $$$$\mathrm{gcf}\left(\mathrm{45},\:\mathrm{60}\right)=\mathrm{3}×\mathrm{5}=\mathrm{15} \\ $$$$\mathrm{gcf}\left(\mathrm{15},\:{x}\right)=\mathrm{3} \\ $$$$\Rightarrow\:{x}=\mathrm{2}^{\mathrm{0}\leqslant{i}} \mathrm{3}^{\mathrm{1}\leqslant{j}} \\ $$$$ \\ $$$$\Rightarrow\:{x}\in\left\{\mathrm{3},\:\mathrm{6},\:\mathrm{9},\:\mathrm{12},\:\mathrm{18},\:\mathrm{36}\right\} \\ $$

Commented by Cheyboy last updated on 02/Aug/18

Thank you sir. I though it was only 3

$${Thank}\:{you}\:{sir}.\:{I}\:{though}\:{it}\:{was}\:{only}\:\mathrm{3} \\ $$

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