Question

A bell rings every 5 seconds. A second bell rings every 6 seconds and a third bell rings every 8 seconds. If all the three rings at the same time at 8:00 a.m., at what time will they all ring together next?
(a) 1 minute past 8:00 a.m.
(b) 2 minutes past 8:00 a.m.
 (c) 3 minutes past 8:00 a.m.
 (d) 4 minutes past 8:00 a.m.

Answer 1

Hint: First, we should know the method to calculate LCM of the numbers as to find the L.C.M. of any numbers we will continuously divide the numbers starting by the smallest prime number which is 2 and divide until we get all numbers as 1. Then, we need to convert the calculated LCM into minutes and add it to the time mentioned at which all the bells ring for the first time which is 8 a.m. Then, on addition, we get the desired time at which all three of them will ring again together.

Complete step-by-step answer:
In this question, we are supposed to find the L.C.M. which is the lowest common multiple of 5 seconds, 6 seconds and 8 seconds to get the common time at which they all will ring together.
To find the L.C.M. of any numbers we will continuously divide the numbers starting by the smallest prime number which is 2 and divide until we get all numbers as 1.
Now, to find the L.C.M. of the 5 seconds, 6 seconds and 8 seconds, we proceed as follows:
$\begin{align}
  & 2\left| \!{\underline {\,
  5,6,8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  5,3,4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  5,3,2 \,}} \right. \\
 & 3\left| \!{\underline {\,
  5,3,1 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5,1,1 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1,1,1 \,}} \right. \\
\end{align}$
Now the L.C.M. of 5 seconds, 6 seconds and 8 seconds is the multiplication of the factors we got on the left side of the calculation.
So, the L.C.M. of 5 seconds, 6 seconds and 8 seconds is:
$2\times 2\times 2\times 3\times 5=120$
The LCM is 120 seconds and we need to convert it into minutes.
Now, we know that:
1 minute=60 seconds
2 minutes=120 seconds
So, we can conclude that all the three rings will ring together after 2 minutes.
So, all the bells will ring together after 8 a.m. at 8.02 a.m. which is 2 minutes past 8 a.m.
Hence, option (b) is correct.

Note: The most common mistake you can attempt while solving this type of the question is in calculating the LCM of the numbers as sometimes we mix the concept of the LCM and HCF. Let us assume two numbers as 4 and 8. The LCM of the two numbers is:
$\begin{align}
  & 2\left| \!{\underline {\,
  4,8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2,4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  1,2 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1,1 \,}} \right. \\
\end{align}$
On multiplication we get $2\times 2\times 2=8$ as LCM.
Bur from the above calculation only HCF is the only factor which is common or we will divide the numbers until any of the numbers becomes 1.
$\begin{align}
  & 2\left| \!{\underline {\,
  4,8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2,4 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1,2 \,}} \right. \\
\end{align}$
So, the HCF of the given two numbers is $2\times 2=4$ which is the common number only.