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Question Number 41534 by mondodotto@gmail.com last updated on 09/Aug/18

Three chidren are playing the game of claping hands,the first child claping hands in every after 1sec,the second child clap hands in every after 10sec and the third child claps in every after 5sec. for how long do all three children will clap their hands together at the same time?

$$\boldsymbol{\mathrm{Three}}\:\boldsymbol{\mathrm{chidren}}\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{playing}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{game}} \\ $$$$\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{claping}}\:\boldsymbol{\mathrm{hands}},\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}}\:\boldsymbol{\mathrm{child}}\:\boldsymbol{\mathrm{claping}}\:\boldsymbol{\mathrm{hands}} \\ $$$$\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{every}}\:\boldsymbol{\mathrm{after}}\:\mathrm{1}\boldsymbol{\mathrm{sec}},\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{second}}\: \\ $$$$\boldsymbol{\mathrm{child}}\:\boldsymbol{\mathrm{clap}}\:\boldsymbol{\mathrm{hands}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{every}}\:\boldsymbol{\mathrm{after}}\:\mathrm{10}\boldsymbol{\mathrm{sec}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{third}}\:\boldsymbol{\mathrm{child}}\:\boldsymbol{\mathrm{claps}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{every}}\:\boldsymbol{\mathrm{after}}\:\mathrm{5}\boldsymbol{\mathrm{sec}}.\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{how}}\:\boldsymbol{\mathrm{long}}\:\boldsymbol{\mathrm{do}}\:\boldsymbol{\mathrm{all}} \\ $$$$\boldsymbol{\mathrm{three}}\:\boldsymbol{\mathrm{children}}\:\boldsymbol{\mathrm{will}}\:\boldsymbol{\mathrm{clap}}\:\boldsymbol{\mathrm{their}}\:\boldsymbol{\mathrm{hands}}\:\boldsymbol{\mathrm{together}}\:\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{same}}\:\boldsymbol{\mathrm{time}}? \\ $$

Answered by alex041103 last updated on 10/Aug/18

Let at time t=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_1 , T_2 and T_3 devide (with zero remainder) T, where T is the smallest, positive number that satisfies those conditions. ⇒T=LCM(T_1 ,T_2 ,T_3 ) , where T_i is the clap period of the i^(th) child. ⇒Ans. T=LCM(1,10,5)=10s

$${Let}\:{at}\:{time}\:{t}=\mathrm{0},\:{all}\:{children}\:{start}\:{clapping}\:{hands}\:{at}\:{the}\:{same}\:{time}. \\ $$$${The}\:{next}\:{time}\:{it}\:{happens}\:{will}\:{be}\:{at}\: \\ $$$${t}={T}.\:{For}\:{this}\:{period}\:{of}\:{time}\:{T},\:{every} \\ $$$${child}\:{will}\:{have}\:{clapped}\:{a}\:{whole}\:{number} \\ $$$${of}\:{times}.\:{That}\:{is}\:{to}\:{say}\:{that}\:{T}_{\mathrm{1}} ,\:{T}_{\mathrm{2}} \:{and} \\ $$$${T}_{\mathrm{3}} \:{devide}\:\left({with}\:{zero}\:{remainder}\right)\:{T},\:{where} \\ $$$${T}\:{is}\:{the}\:{smallest},\:{positive}\:{number}\:{that}\:{satisfies} \\ $$$${those}\:{conditions}. \\ $$$$\Rightarrow{T}={LCM}\left({T}_{\mathrm{1}} ,{T}_{\mathrm{2}} ,{T}_{\mathrm{3}} \right)\:,\:{where}\:{T}_{{i}} \:{is} \\ $$$${the}\:{clap}\:{period}\:{of}\:{the}\:{i}^{{th}} \:{child}. \\ $$$$\Rightarrow{Ans}.\:{T}={LCM}\left(\mathrm{1},\mathrm{10},\mathrm{5}\right)=\mathrm{10}{s} \\ $$

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