Question

Four bells ring at intervals of 6, 7, 8 and 9 seconds respectively. All the bells ring together after ---seconds.\[\]
A.504\[\]
B.516\[\]
C.508\[\]
D.512\[\]

Answer 1

Hint: We see that the first, second, third and fourth bell will ring when multiples of 6 .7 .8 and 9 seconds have passed from the time of ringing of the first bell . They are going to ring together when the multiples of 6, 7, 8 and 9 seconds are the same. So the first time they will ring together is when the least common multiple (LCM) of 6, 7, 8 and 9 seconds have passed. We use tabular method to find LCM.\[\]

Complete step-by-step answer:
We know that the least common multiple (LCM) of two numbers $a,b$ is the common multiple of both $a,b$ which is least.
We are given that four bells ring at intervals of 6, 7, 8 and 9 seconds respectively. Let the time when the first bell rings is H. \[\]
So the first bell will ring at interval of 6 seconds which means it will ring at 6, 12, 18... seconds passed from time H.\[\]
The second bell will ring at interval of 7 seconds which means it will ring at 7, 14, 21... seconds passed from time H. \[\]
The third bell will ring at interval of 8 seconds which means it will ring at 8, 16, 24... seconds from passed time H.\[\]
The fourth bell will ring at interval of 9 seconds which means it will ring at 9, 18, 27... seconds passed from time H.\[\]
So the first, second, third and fourth bells will ring at all multiples of 6, 7, 8 and 9 seconds passed from time H. They are going to ring together when the multiples of 6, 7, 8 and 9 seconds are the same. So the first time they will ring together is when the least common multiple of 6, 7, 8 and 9 seconds have passed. We use tabular method to find LCM
\[\begin{align}
  & 2\left| \!{\underline {\,
  6,7,8,9 \,}} \right. \\
 & 2\left| \!{\underline {\,
  3,7,4,9 \,}} \right. \\
 & \text{3}\left| \!{\underline {\,
  \text{3,7,2,9} \,}} \right. \\
 & \hspace{0.3 cm }1,7,2,3 \\
 & \text{ } \\
\end{align}\]
So the LCM is $2\times 3\times 7\times 2\times 2\times 3=504$. So the bells ring after 504 seconds and the correct option is C.\[\]

So, the correct answer is “Option C”.

Note: We can alternatively find LCM using the prime factorization of 6, 7, 8 and 9 where we choose numbers with highest occurring prime factors. We can convert the 504 seconds to minutes dividing by 60 and find 8 minutes 24 seconds. We can also find the number of times they ring together by one hour by dividing $1\text{ hour}=3600\text{ seconds}$ by 504.