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?


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

$Three chidren are playing the game$ $of claping hands, the first child claping hands$ $in every after 1 \sec, the second$ $child clap hands in every after 10 \sec and the$ $third child claps in every after 5 \sec . for how long do all$ $three children will clap their hands together at the same time ?$
	Answered by alex041103 last updated on 10/Aug/18

	$L e t a t t i m e t = 0, a l l c h i l d r e n s t a r t c l a p p i n g h a n d s a t t h e s a m e t i m e .$ $T h e n e x t t i m e i t h a p p e n s w i l l b e a t$ $t = T . F o r t h i s p e r i o d o f t i m e T, e v e r y$ $c h i l d w i l l h a v e c l a p p e d a w h o l e n u m b e r$ $o f t i m e s . T h a t i s t o s a y t h a t T_{1}, T_{2} a n d$ $T_{3} d e v i d e (w i t h z e r o r e m a i n d e r) T, w h e r e$ $T i s t h e s m a l l e s t, p o s i t i v e n u m b e r t h a t s a t i s f i e s$ $t h o s e c o n d i t i o n s .$ $\Rightarrow T = L C M (T_{1}, T_{2}, T_{3}), w h e r e T_{i} i s$ $t h e c l a p p e r i o d o f t h e i^{t h} c h i l d .$ $\Rightarrow A n s . T = L C M (1, 10, 5) = 10 s$
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