Question Number 104536 by Anindita last updated on 22/Jul/20
$$\boldsymbol{\mathrm{The}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{two}}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{is}}\:\mathrm{384}\:\boldsymbol{\mathrm{and}} \\ $$$$\mathrm{t}\boldsymbol{\mathrm{heir}}\:\boldsymbol{\mathrm{GCD}}\:\boldsymbol{\mathrm{is}}\:\mathrm{48}.\:\boldsymbol{\mathrm{What}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{LCM}}\:\boldsymbol{\mathrm{of}} \\ $$$$\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{numbers}}\:? \\ $$
Commented by 1549442205PVT last updated on 22/Jul/20
$$\mathrm{Similar}\:\mathrm{to}\:\mathrm{Q104413} \\ $$
Answered by mr W last updated on 22/Jul/20
$${gcd}\left({x},{y}\right)=\mathrm{48} \\ $$$$\Rightarrow{x}=\mathrm{48}{m} \\ $$$$\Rightarrow{y}=\mathrm{48}{n} \\ $$$${with}\:{gcd}\left({m},{n}\right)=\mathrm{1}\:\:\:…\left({i}\right) \\ $$$${x}+{y}=\mathrm{48}\left({m}+{n}\right)=\mathrm{384} \\ $$$$\Rightarrow{m}+{n}=\mathrm{8}\:\:\:…\left({ii}\right) \\ $$$${lcm}\left({x},{y}\right)=\mathrm{48}×{lcm}\left({m},{n}\right) \\ $$$$ \\ $$$${case}\:\mathrm{1}:\:{m}=\mathrm{1},{n}=\mathrm{7}\:{or}\:{m}=\mathrm{7},\:{n}=\mathrm{1} \\ $$$${lcm}\left({x},{y}\right)=\mathrm{48}×{lcm}\left(\mathrm{1},\mathrm{7}\right)=\mathrm{336} \\ $$$${case}\:\mathrm{2}:\:{m}=\mathrm{3},{n}=\mathrm{5}\:{or}\:{m}=\mathrm{5},\:{n}=\mathrm{3} \\ $$$${lcm}\left({x},{y}\right)=\mathrm{48}×{lcm}\left(\mathrm{3},\mathrm{5}\right)=\mathrm{720} \\ $$$$ \\ $$$${so}\:{the}\:{answer}\:{is}\:\mathrm{336}\:{or}\:\mathrm{720}. \\ $$
Commented by Rasheed.Sindhi last updated on 22/Jul/20
$$\mathcal{G}\overset{\mathcal{R}} {}\mathcal{E}\overset{\mathcal{A}} {}\mathcal{T}\:\:! \\ $$