Question Number 41534 by mondodotto@gmail.com last updated on 09/Aug/18
$$\boldsymbol{\mathrm{Three}}\:\boldsymbol{\mathrm{chidren}}\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{playing}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{game}} \\ $$$$\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{claping}}\:\boldsymbol{\mathrm{hands}},\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}}\:\boldsymbol{\mathrm{child}}\:\boldsymbol{\mathrm{claping}}\:\boldsymbol{\mathrm{hands}} \\ $$$$\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{every}}\:\boldsymbol{\mathrm{after}}\:\mathrm{1}\boldsymbol{\mathrm{sec}},\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{second}}\: \\ $$$$\boldsymbol{\mathrm{child}}\:\boldsymbol{\mathrm{clap}}\:\boldsymbol{\mathrm{hands}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{every}}\:\boldsymbol{\mathrm{after}}\:\mathrm{10}\boldsymbol{\mathrm{sec}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{third}}\:\boldsymbol{\mathrm{child}}\:\boldsymbol{\mathrm{claps}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{every}}\:\boldsymbol{\mathrm{after}}\:\mathrm{5}\boldsymbol{\mathrm{sec}}.\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{how}}\:\boldsymbol{\mathrm{long}}\:\boldsymbol{\mathrm{do}}\:\boldsymbol{\mathrm{all}} \\ $$$$\boldsymbol{\mathrm{three}}\:\boldsymbol{\mathrm{children}}\:\boldsymbol{\mathrm{will}}\:\boldsymbol{\mathrm{clap}}\:\boldsymbol{\mathrm{their}}\:\boldsymbol{\mathrm{hands}}\:\boldsymbol{\mathrm{together}}\:\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{same}}\:\boldsymbol{\mathrm{time}}? \\ $$
Answered by alex041103 last updated on 10/Aug/18
$${Let}\:{at}\:{time}\:{t}=\mathrm{0},\:{all}\:{children}\:{start}\:{clapping}\:{hands}\:{at}\:{the}\:{same}\:{time}. \\ $$$${The}\:{next}\:{time}\:{it}\:{happens}\:{will}\:{be}\:{at}\: \\ $$$${t}={T}.\:{For}\:{this}\:{period}\:{of}\:{time}\:{T},\:{every} \\ $$$${child}\:{will}\:{have}\:{clapped}\:{a}\:{whole}\:{number} \\ $$$${of}\:{times}.\:{That}\:{is}\:{to}\:{say}\:{that}\:{T}_{\mathrm{1}} ,\:{T}_{\mathrm{2}} \:{and} \\ $$$${T}_{\mathrm{3}} \:{devide}\:\left({with}\:{zero}\:{remainder}\right)\:{T},\:{where} \\ $$$${T}\:{is}\:{the}\:{smallest},\:{positive}\:{number}\:{that}\:{satisfies} \\ $$$${those}\:{conditions}. \\ $$$$\Rightarrow{T}={LCM}\left({T}_{\mathrm{1}} ,{T}_{\mathrm{2}} ,{T}_{\mathrm{3}} \right)\:,\:{where}\:{T}_{{i}} \:{is} \\ $$$${the}\:{clap}\:{period}\:{of}\:{the}\:{i}^{{th}} \:{child}. \\ $$$$\Rightarrow{Ans}.\:{T}={LCM}\left(\mathrm{1},\mathrm{10},\mathrm{5}\right)=\mathrm{10}{s} \\ $$