Question Number 133772 by mathocean1 last updated on 24/Feb/21
$${Three}\:{devices}\:{A};\:{B}\:;{C}\:{shine}\:{like}\:{that}: \\ $$$${A}\:{shines}\:{every}\:\mathrm{25}\:{minutes} \\ $$$${B}\:{shines}\:{every}\:\mathrm{30}\:{minutes} \\ $$$${C}\:{shines}\:{every}\:\mathrm{35}\:{minutes}. \\ $$$${The}\:{three}\:{devices}\:{shined}\:{simultaneous}, \\ $$$$\left({at}\:{the}\:{same}\:{time}\right)\:{at}\:\mathrm{10}\:{pm}. \\ $$$${At}\:{which}\:{time}\:{will}\:{they}\:{shine}\:{simultaneous}\: \\ $$$${for}\:{the}\:{first}\:{time}\:{after}\:{midnight}? \\ $$
Answered by TheSupreme last updated on 24/Feb/21
$${N}={k}_{\mathrm{1}} \mathrm{5}^{\mathrm{2}} ={k}_{\mathrm{2}} \mathrm{6}×\mathrm{5}={k}_{\mathrm{3}} \mathrm{7}×\mathrm{5} \\ $$$${N}_{{min}} ={mcm}\left(\mathrm{5}^{\mathrm{2}} ,\mathrm{3}×\mathrm{2},\mathrm{7}\right)/\mathrm{60}=\frac{\mathrm{5}^{\mathrm{2}} ×\mathrm{3}×\mathrm{2}×\mathrm{7}}{\mathrm{3}×\mathrm{2}^{\mathrm{2}} ×\mathrm{5}}=\mathrm{35}/\mathrm{2}=\mathrm{17}.\mathrm{5}{h} \\ $$$$\mathrm{10}:\mathrm{00}{pm} \\ $$$$\mathrm{27}.\mathrm{30}{pm} \\ $$$$\mathrm{3}.\mathrm{30}{pm} \\ $$$$ \\ $$