Question

Three bells ring at intervals of $3,5,$ and $7$ seconds, respectively. If they ring together, how many times do they ring together in the next $7$ minutes?

• A. $7$
• B. $10$
• C. $4$
• D. $3$

Answer 1

The correct option is C $4$

If the first bell rings after $3$ seconds, it will again ring after $6,9,12,\mathrm{15...}$ seconds.
Similarly, other bells will ring after $5,10,15,...$ seconds and $7,14,21,\mathrm{28..}$ seconds, respectively.

But, if we consider ringing first and second bells together, they will ring together after $15$ seconds.
$15$ is the LCM of both $3$ and $5$.

Similarly, the time after which all the bells will ring together is the LCM of interval durations of all the bells.

Since the timings of the bells are co-prime numbers, LCM is the product of the timing of each bell.
$\Rightarrow$ LCM of $3,5$ and $7$ is $3\times 5\times 7=105$

At ${105}^{th}$ second, they will ring together.

Now, number of times these bell ring together in $7$ minutes ($420$ seconds) $=\frac{7\times 60}{105}=\frac{420}{105}=4\text{times}$.