The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change simultaneously again?

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 do :

We have to find at what time three signals change together again.

Solution :

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

Therefore,

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

Prime factorization of 48, 72, and 108 are

$48 = 2\times 2\times 2\times 2\times 3$

$72 = 2\times 2\times 2\times 3\times 3$

$108 = 2\times 2\times 3\times 3\times 3$

LCM of 48, 72 and 108 $= 2\times 2\times 2\times 2\times 3\times 3\times 3 = 432$.

This implies,

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

$432\ seconds = (7\times 60 + 12)\ seconds = 6\ minutes 12\ seconds$.

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