Three bells toll at intervals of $9$, $12$ and $15$ minutes respectively. If they all toll together at $8:00$ a.m., at what time will they next toll together?

Step-by-step solution

1. The bells toll together again after a number of minutes that is a multiple of all three intervals, so we need the LCM of $9$, $12$ and $15$.
2. Write each interval as a product of primes: $9=3^2$, $12=2^2\times 3$, and $15=3\times 5$.
3. For the LCM, take each prime to its highest power: $2^2$, $3^2$ and $5$.
4. LCM $=2^2\times 3^2\times 5=4\times 9\times 5=180$ minutes.
5. Convert to hours: $180$ minutes $=3$ hours.
6. Adding $3$ hours to $8:00$ a.m. gives $11:00$ a.m.

Concept tested: Using the LCM for repeating events

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