Four bells ring at 16, 24, 36, 42 minute intervals. If they ring together at 6.00 am, after bow long would they next ring together?

Options

A. 842 minutes
B. 964 minutes
C. 886 minutes
D. 1008 minutes

Answer

To solve the problem of when the four bells will next ring together after ringing together at 6:00 AM, we need to find the least common multiple (LCM) of the intervals at which they ring: 16, 24, 36, and 42 minutes.

### Step-by-Step Solution:

1. **Identify the intervals**: The bells ring at intervals of 16, 24, 36, and 42 minutes.

2. **Find the prime factorization of each interval**:
- **16**: \(16 = 2^4\)
- **24**: \(24 = 2^3 \times 3^1\)
- **36**: \(36 = 2^2 \times 3^2\)
- **42**: \(42 = 2^1 \times 3^1 \times 7^1\)

3. **Determine the LCM**:
- For the LCM, we take the highest power of each prime factor from the factorizations:
- For \(2\): The highest power is \(2^4\) (from 16).
- For \(3\): The highest power is \(3^2\) (from 36).
- For \(7\): The highest power is \(7^1\) (from 42).
- Therefore, the LCM is:
\[ LCM = 2^4 \times 3^2 \times 7^1 \]

4. **Calculate the LCM**:
- First, calculate \(2^4 = 16\).
- Next, calculate \(3^2 = 9\).
- Now, multiply these results with \(7\):
\[ LCM = 16 \times 9 \times 7 \]
- Calculate \(16 \times 9 = 144\).
- Finally, calculate \(144 \times 7 = 1008\).

5. **Interpret the result**:
- The LCM of 16, 24, 36, and 42 is 1008 minutes.
- To find out how long after 6:00 AM they will next ring together, we convert 1008 minutes into hours and minutes:
- \(1008 \div 60 = 16\) hours with a remainder of \(48\) minutes.
- Therefore, they will ring together again at \(6:00 AM + 16\) hours and \(48\) minutes, which is \(10:48 PM\).

### Final Answer:
The bells will next ring together after **1008 minutes**, which is **16 hours and 48 minutes** after 6:00 AM.

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