At what kind of situations LCM has to be used ?

Given :

Traffic lights at three different road crossing change after every 48 seconds, 72 seconds and 108 seconds respectively.

They changed simultaneously at 7 AM.

To find :

We have to find the LCM of 48, 72 and 108

Solution:

The three lights change simultaneously on the common multiples of all the three.

Therefore,

Prime factorisation of 48, 72 and 108 are

$48 = 2\times 2\times2\times2\times3$

$72 = 2\times2\times2\times3\times3$

$108 = 2\times2\times3\times3\times3$

$LCM of 48, 72 and 108 = 2\times2\times2\times2\times3\times3\times3 = 432.$

This implies,

The three lights change simultaneously after a minimum of 432 seconds.

$432 seconds = (7\times60 + 12) seconds = 6 minutes 12 seconds.$

The lights will change simultaneously after 7 AM at 6 minutes 12 seconds after 7.