Question

Five bells begin to ring together and ring respectively at intervals of 6, 5, 7, 10 and 12 seconds. How many times will they ring together in one hour excluding the one at the start?

• A. 7 times
• B. 8 times
• C. 9 times
• D. 11 times

Hint:

When multiple periodic events start together, their next simultaneous occurrence time is the LCM of their periods. In a time window, count the multiples of that LCM.

Answer 1

The Correct Option is B

Step 1: Find the LCM of the intervals.
\[ \mathrm{LCM}(6,5,7,10,12)=\mathrm{LCM}(2\cdot3,\ 5,\ 7,\ 2\cdot5,\ 2^2\cdot3)=2^2\cdot3\cdot5\cdot7=420\ \text{s}. \] Step 2: Count how many multiples of 420 s occur in one hour.
One hour \(=3600\) s. The bells coincide at \(t=420, 840, \ldots\) up to \( \le 3600\).
Number of coincidences (excluding \(t=0\)) is \[ \left\lfloor \frac{3600}{420} \right\rfloor = 8. \] \[ \boxed{8\ \text{times}} \]