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-f

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