Two alarm clock ring their alarms at regular intervals of 50 seconds and 48 seconds. If they first beep together at 12 noon, at what time will they beep again?

Options

A. 12: 20 p.m.
B. 01 : 05 p.m.
C. 02 : 20 p.m.
D. 12 : 35 p.m.

Answer

To solve the problem of when the two alarm clocks will beep together again after first beeping together at 12 noon, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the intervals of the alarms**:
- The first alarm clock rings every 50 seconds.
- The second alarm clock rings every 48 seconds.

2. **Find the Least Common Multiple (LCM)**:
- To determine when both alarms will beep together again, we need to find the LCM of 50 and 48.
- We can find the LCM by using the prime factorization method.

3. **Prime Factorization**:
- The prime factorization of 50 is \(2^1 \times 5^2\).
- The prime factorization of 48 is \(2^4 \times 3^1\).

4. **Calculate the LCM**:
- The LCM is found by taking the highest power of each prime factor from both numbers.
- For \(2\), the highest power is \(2^4\).
- For \(3\), the highest power is \(3^1\).
- For \(5\), the highest power is \(5^2\).
- Therefore, the LCM is \(2^4 \times 3^1 \times 5^2\).

5. **Calculate the LCM step-by-step**:
- \(2^4 = 16\)
- \(3^1 = 3\)
- \(5^2 = 25\)
- Now multiply these together:
\[ LCM = 16 \times 3 \times 25 \]
- First, calculate \(16 \times 3 = 48\).
- Then, calculate \(48 \times 25 = 1200\).

6. **Convert seconds to minutes**:
- Since the LCM is 1200 seconds, we need to convert this into minutes.
- We know that 1 minute = 60 seconds.
- Therefore, \(1200 \text{ seconds} = \frac{1200}{60} \text{ minutes} = 20 \text{ minutes}\).

7. **Determine the time they beep together again**:
- They first beep together at 12:00 PM (noon).
- Adding 20 minutes to 12:00 PM gives us 12:20 PM.

### Final Answer:
The two alarm clocks will beep together again at **12:20 PM**.