Question Number 41167 by Cheyboy last updated on 02/Aug/18
$${The}\:{LCM}\:{and}\:{the}\:{GCF}\:{of}\:{three} \\ $$$${intergers}\:{are}\:\mathrm{180}\:{and}\:\mathrm{3}\:{respectively}. \\ $$$${Two}\:{numbers}\:{are}\:\mathrm{45}\:{and}\:\mathrm{60}. \\ $$$${What}\:{is}\:{the}\:{third}\:{number} \\ $$
Answered by MJS last updated on 02/Aug/18
$$\mathrm{45}=\mathrm{3}^{\mathrm{2}} \mathrm{5} \\ $$$$\mathrm{60}=\mathrm{2}^{\mathrm{2}} \mathrm{3}×\mathrm{5} \\ $$$$\mathrm{lcm}\left(\mathrm{45},\:\mathrm{60}\right)=\mathrm{2}^{\mathrm{2}} \mathrm{3}^{\mathrm{2}} \mathrm{5}=\mathrm{180} \\ $$$$\mathrm{lcm}\left(\mathrm{180},\:{x}\right)=\mathrm{180} \\ $$$$\Rightarrow\:{x}=\mathrm{2}^{\mathrm{0}\leqslant{i}\leqslant\mathrm{2}} \mathrm{3}^{\mathrm{0}\leqslant{j}\leqslant\mathrm{2}} \mathrm{5}^{\mathrm{0}\leqslant{k}\leqslant\mathrm{1}} \\ $$$$\mathrm{gcf}\left(\mathrm{45},\:\mathrm{60}\right)=\mathrm{3}×\mathrm{5}=\mathrm{15} \\ $$$$\mathrm{gcf}\left(\mathrm{15},\:{x}\right)=\mathrm{3} \\ $$$$\Rightarrow\:{x}=\mathrm{2}^{\mathrm{0}\leqslant{i}} \mathrm{3}^{\mathrm{1}\leqslant{j}} \\ $$$$ \\ $$$$\Rightarrow\:{x}\in\left\{\mathrm{3},\:\mathrm{6},\:\mathrm{9},\:\mathrm{12},\:\mathrm{18},\:\mathrm{36}\right\} \\ $$
Commented by Cheyboy last updated on 02/Aug/18
$${Thank}\:{you}\:{sir}.\:{I}\:{though}\:{it}\:{was}\:{only}\:\mathrm{3} \\ $$