Three lightships flash simultaneously at 6:00 a.m. The first lightship flashes every 12 seconds, the second lightship flashes after every 30 seconds, and the third lightship every 66 seconds. At what time will the three lightships next flash together?
(A) 6:09 am
(B) 6:10 am
(C) 6:11 am
(D) 6:12 am

Hint: The time duration for the flash of first lightship, second lightship, and the third lightship are 12 seconds, 30 seconds, and 66 seconds respectively. To find the time at which all three lightships will again flash simultaneously, use the logic that the lowest multiple of the time intervals of flash will be the time at which all three lightships will again flash simultaneously. Now, use this logic and proceed further.

Complete step-by-step answer:
According to the question, we are given,
The time duration for the flash of first lightship = 12 seconds ……………………………………………..(1)
The time duration for the flash of second lightship = 30 seconds ……………………………………………..(2)
The time duration for the flash of third lightship = 66 seconds ……………………………………………..(3)
It is also given that all three lightships flashed simultaneously at 6:00 a.m. for the first time ………………………………………………(4)
We have to find the time at which all three lightships will again flash simultaneously.
We can think of the logic that the lowest multiple of the time intervals of flash will be the time at which all three lightships will again flash simultaneously ………………………………………….(5)
Using this logic, we need the lowest common factor of the time intervals for the flash of the first lightship, second lightship, and the third lightship that are 12 seconds, 30 seconds, and 66 seconds.
Now, finding the LCM of 12, 30, and 66,
\[\begin{align}
  & 6\left| \!{\underline {\,
  12,30,66 \,}} \right. \\
 & \,\,\left| \!{\underline {\,
  2,5,11 \,}} \right. \\
\end{align}\]
On multiplying, we get, \[6\times 2\times \,5\times 11=660\] ……………………………………(6)
Now, from equation (5) and equation (6), we can say that after 660 seconds from 6:00 a.m., all three lightships will be again flashing simultaneously.
We also know the relation, \[\text{60}\,\text{seconds=1 minutes}\Rightarrow \text{1}\,\text{seconds=}\dfrac{\text{1}}{\text{60}}\,\text{minutes}\] ……………………………………………...(7)
Now, on converting 660 seconds into minutes, we get
\[\begin{align}
  & \text{=660 seconds} \\
 & \text{=660}\times \dfrac{1}{60}\,\text{minutes} \\
 & \text{=11 minutes} \\
\end{align}\]
So, after 11 minutes from the start, all three lightships will be again flashing simultaneously …………………………………….(8)
From equation (4) and equation (8), we can say that at 6:11 a.m. all three lightships will be again flashing simultaneously.

So, the correct answer is “Option (C)”.

Note:Whenever this type of question is asked where we are given some time intervals and we have to find the common time. Always approach this question by taking the LCM of the time intervals and proceed further.