Finding the Next Simultaneous Bell Ring Time

The problem asks for the next time three bells, ringing at intervals of 15, 30, and 45 minutes, will ring together. They last rang together at 8:00 AM.

Calculating the Least Common Multiple (LCM)

To find when the bells will ring together again, we need to find the Least Common Multiple (LCM) of their ringing intervals (15, 30, and 45 minutes).

1. Prime Factorization:
   • $15 = 3 \times 5$
   • $30 = 2 \times 3 \times 5$
   • $45 = 3^2 \times 5$
2. LCM Calculation: Take the highest power of each prime factor present in any factorization.
   • The prime factors are 2, 3, and 5.
   • Highest power of 2 is $2^1$.
   • Highest power of 3 is $3^2$.
   • Highest power of 5 is $5^1$.
   • Therefore, $LCM(15, 30, 45) = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90$ minutes.

The bells will ring together every 90 minutes.

Determining the Next Ring Time

The bells rang together at 8:00 AM. They will ring together again after 90 minutes.

• Convert 90 minutes to hours and minutes: $90 \text{ minutes} = 1 \text{ hour and } 30 \text{ minutes}$.
• Add this duration to the initial time: $8:00 \text{ AM} + 1 \text{ hour } 30 \text{ minutes} = 9:30 \text{ AM}$.

Thus, the bells will ring together again at 9:30 AM.