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 

= \(\frac{(Total\, distance\, of\, the\, circular\, field\,) }{ (distance\, covered\, in\, 1\, day\, by\, a\, cyclist\,)}\)

So, for the 1st cyclist, number of days = \(\frac{360}{48}\) = 7.5 which is = 180 hours [∵1 day = 24 hours] 

2nd cyclist, number of days = \(\frac{360}{60}\) = 6 which is = 144 hours 

3rd 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 

180 = 2² x 3² x 5 

144 = 2⁴ x 3² 

120 = 2³ x 3 x 5 

⇒ L.C.M (180, 144 and 120) = 2⁴ x 3² x 5 = 720 

So, this means that after 720 hours the three cyclists meet again. 

⇒ 720 hours = \(\frac{720}{24}\) = 30 days [∵1 day = 24 hours] 

Thus, all the three cyclists will meet again after 30 days.