A bell rings every 5 seconds. A second bell rings every 6 seconds and a third one rings every 8 seconds. If all the three rings at the same time at 8.00 a.m., at what time will they all ring together next?

Options

A. 1 minute past 8.00 a.m.
B. 2 minutes past 8.00 a.m.
C. 3 minutes past 8.00 a.m.
D. 4 minutes past 8.00 a.m.

Answer

To solve the problem of when the three bells will ring together next after 8:00 a.m., we need to find the Least Common Multiple (LCM) of the ringing intervals of the bells, which are 5 seconds, 6 seconds, and 8 seconds.

### Step-by-Step Solution:

1. **Identify the ringing intervals**:
- First bell rings every 5 seconds.
- Second bell rings every 6 seconds.
- Third bell rings every 8 seconds.

2. **Find the LCM of the intervals**:
- The intervals are 5, 6, and 8.
- We can find the LCM by using the prime factorization method.

- **Prime Factorization**:
- 5 is a prime number: \(5 = 5^1\)
- 6 can be factored into \(2 \times 3\): \(6 = 2^1 \times 3^1\)
- 8 can be factored into \(2 \times 2 \times 2\): \(8 = 2^3\)

- **Taking the highest powers of all prime factors**:
- For 2: the highest power is \(2^3\) (from 8).
- For 3: the highest power is \(3^1\) (from 6).
- For 5: the highest power is \(5^1\) (from 5).

- **Calculating the LCM**:
\[ \text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 \]
\[ 8 \times 3 = 24 \]
\[ 24 \times 5 = 120 \]
- Therefore, the LCM of 5, 6, and 8 is **120 seconds**.

3. **Convert seconds into minutes**:
- Since 120 seconds is equal to \( \frac{120}{60} = 2 \) minutes.

4. **Determine the next ringing time**:
- The bells all ring together at 8:00 a.m. and will ring together again after 2 minutes.
- Adding 2 minutes to 8:00 a.m. gives us:
\[ 8:00 \text{ a.m.} + 2 \text{ minutes} = 8:02 \text{ a.m.} \]

### Final Answer:
The three bells will all ring together next at **8:02 a.m.**.

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