In order to calculate the time take before they meet again, we must first find out the individual time taken by each cyclist in covering the total distance.
Number of days a cyclist takes to cover the circular field
\(=\frac{(\text { Total distance of the circular ficld })}{\text { (disiance coecred in } 1 \text { day by a cyclist) }}\)
So, for the 1st cyclist, number of days \(=\frac{360}{48}=7.5\) which is \(=180\) hours \([\because 1\) day \(=24\) hours]
2nd cyclist, number of days \(=\frac{360}{60}=6\) which is \(=144\) hours
3 rd cyclist, number of days \(=\frac{360}{72}=5\) which is 120 hours
Now, by finding the LCM \((180,144\) and 120\()\) we'll get to know after how many hours the three cyclists meet again.
By prime factorisation, we get
$$
\begin{aligned}
&180=2^{2} \times 3^{2} \times 5 \\
&144=2^{4} \times 3^{2} \\
&120=2^{3} \times 3 \times 5 \\
&=\text { L.C.M }(180,144 \text { and } 120)=2^{4} \times 3^{2} \times 5=720
\end{aligned}
$$
So, this means that after 720 hours the three cyclists meet again.
\(=720\) hours \(=\frac{720}{24}=30\) days \([\because 1\) day \(=24\) hours \(]\)
Thus, all the three cyclists will meet again after 30 days.
