Question

Four mobile phones commence vibrating together and vibrate at an interval of 16 secs, 9 secs, 8 secs and 4 secs respectively. In 12 minutes how many times will they vibrate together?
A. 7
B. 4
C.8
D. none of these

Answer 1

Hint:
We have been given that four mobile phones vibrate at an interval of 16 secs, 9 secs, 8 secs, and 4 secs respectively. They will keep vibrating to the multiple of those numbers. So, we find the L.C.M of the numbers to find their common vibrating interval. Then we divide the total time by the L.C.M to find the solution.

Complete step by step answer:
Four mobile phones commence vibrating together and vibrate at an interval of 16 secs, 9 secs, 8 secs, and 4 secs respectively.
We need to find the L.C.M of the numbers 16, 9, 8, 4. They will keep vibrating to the multiple of those numbers. So, they will vibrate together when their multiples are the same.
So, we are trying to find the least common multiple number of those given inputs.
\[\begin{align}
  & 2\left| \!{\underline {\,
  16,9,8,4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8,9,4,2 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4,9,2,1 \,}} \right. \\
 & 1\left| \!{\underline {\,
  2,9,1,1 \,}} \right. \\
\end{align}\]
So, the L.C.M of the numbers 16, 9, 8, 4 is $ {{2}^{4}}\times 9=144 $ . This means after every 144 seconds they will vibrate together.
We need to find the number of times they will vibrate together in 12 minutes.
12 minutes is equal to $ 12\times 60=720 $ seconds.
We divide 720 by 144 to find the number of times they will vibrate together.
The number of times is $ \dfrac{720}{144}=5 $ .
The correct option is D.

Note:
If the total number of vibrating times is asked then we need to add 1 to the final answer as the phones started vibrating together at the very start. We have to count that in those 12 minutes.