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Real Numbers — Maths STD 10 — Question

Maharashtra BoardEnglish MediumSTD 10MathsReal Numbers4 Marks
Question
The traffic lights at three different road crossing change after every $48$ secons, $72$ seconds and $108$ seconds respectively. If they all change simultaneously at $8 a.m.$ then at what time will they again change simultaneously?
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Answer

Let us find the LCM of $48, 72$ and $108$ through prime factorisation:
$\begin{array}{c|c} 2 & 48 \\ \hline 2 & 24\\ \hline2&12\\ \hline2&6\\ \hline&3 \end{array}$ $\begin{array}{c|c} 2 & 72 \\ \hline 2 & 36\\ \hline2&18\\ \hline3&9\\ \hline&3 \end{array}$ $\begin{array}{c|c} 2 & 108 \\ \hline 2 & 54\\ \hline3&27\\ \hline3&9\\ \hline&3 \end{array}$
$48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3$
$72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$
$108 = 2 \times 2 \times 3 \times 3 \times 3 = 2^2 \times 3^3$
LCM of $48, 72, 108$ is $2^4 \times 3^3$
$= 16 \times 27\ sec$
$= 432\ sec$
$=7$ min $12$ sec
Three bells toll together after $7$ min $12$ sec.

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