How to Solve the 'Four Bells Toll Together' Problem

We need to find a future time when all four bells will ring at the exact same moment.

• To happen at the same moment, the number of seconds that pass must be a perfect multiple of 7, a multiple of 8, a multiple of 11, and a multiple of 12.
• Therefore, we need to find the Least Common Multiple (LCM) of these four numbers.

Let's find the LCM of 7, 8, 11, and 12.

• Since 7 and 11 are prime numbers, they cannot be broken down further.
• 8 and 12 share a common factor of 4.
• Using the prime factorization or division method:
  • LCM = $4 \times 2 \times 3 \times 7 \times 11$
  • Let's multiply: $8 \times 3 = 24$. Then $24 \times 7 = 168$. Then $168 \times 11 = 1848$.
• The LCM is 1,848.
• This means all four bells will ring together exactly 1,848 seconds after 9:00 AM.

1,848 seconds is confusing, so let's convert it into minutes and seconds by dividing by 60.

• $1848 \div 60 = 30$ minutes with a remainder of 48 seconds.
• (Because $30 \times 60 = 1800$. And $1848 - 1800 = 48$).
• So, they will ring together after 30 minutes and 48 seconds.

If they first rang together exactly at 9:00:00 AM, you simply add the time. They will toll together again at exactly 9:30:48 AM.