Four different bells beep after every 20 minutes, 40 minutes, 1 hour 20 minutes and 1 hour 40 minutes. If all bell rings (beeped) together at 5:00 a.m then they will again beep together at

Options

A. `6:00` p.m.
B. `11:40` a.m
C. `12:00` noon
D. `9:0` p.m

Answer

To solve the problem of when the four bells will beep together again after starting at 5:00 a.m., we need to follow these steps:

### Step-by-Step Solution:

1. **Convert all time intervals to minutes:**
- The first bell beeps every 20 minutes.
- The second bell beeps every 40 minutes.
- The third bell beeps every 1 hour 20 minutes, which is \(60 + 20 = 80\) minutes.
- The fourth bell beeps every 1 hour 40 minutes, which is \(60 + 40 = 100\) minutes.

2. **List the intervals in minutes:**
- Bell 1: 20 minutes
- Bell 2: 40 minutes
- Bell 3: 80 minutes
- Bell 4: 100 minutes

3. **Find the Least Common Multiple (LCM) of these intervals:**
- Factor each interval into its prime factors:
- 20 = \(2^2 \times 5^1\)
- 40 = \(2^3 \times 5^1\)
- 80 = \(2^4 \times 5^1\)
- 100 = \(2^2 \times 5^2\)

4. **Determine the highest powers of each prime factor:**
- For \(2\): The highest power is \(2^4\) (from 80).
- For \(5\): The highest power is \(5^2\) (from 100).

5. **Calculate the LCM:**
- LCM = \(2^4 \times 5^2 = 16 \times 25 = 400\) minutes.

6. **Convert 400 minutes into hours and minutes:**
- 400 minutes = \(6\) hours and \(40\) minutes.

7. **Add this time to the initial time of 5:00 a.m.:**
- 5:00 a.m. + 6 hours and 40 minutes = 11:40 a.m.

### Final Answer:
The bells will beep together again at **11:40 a.m.**

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