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 at 12 noon, we need to find the least common multiple (LCM) of their ringing intervals, which are 50 seconds and 48 seconds.

### Step-by-Step Solution:

1. **Identify the intervals**: The two alarm clocks ring at intervals of 50 seconds and 48 seconds.

2. **Find the prime factorization of each interval**:
- For 50:
- 50 = 2 × 25
- 25 = 5 × 5
- So, the prime factorization of 50 is: \( 2^1 \times 5^2 \)
- For 48:
- 48 = 2 × 24
- 24 = 2 × 12
- 12 = 2 × 6
- 6 = 2 × 3
- So, the prime factorization of 48 is: \( 2^4 \times 3^1 \)

3. **Determine the LCM**:
- To find the LCM, take the highest power of each prime factor from both factorizations:
- For 2: The highest power is \( 2^4 \) (from 48).
- For 3: The highest power is \( 3^1 \) (from 48).
- For 5: The highest power is \( 5^2 \) (from 50).
- Therefore, the LCM is:
\[ LCM = 2^4 \times 3^1 \times 5^2 \]

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

5. **Convert LCM to time**:
- The LCM of 50 seconds and 48 seconds is 1200 seconds.
- Now, convert 1200 seconds into minutes:
\[ 1200 \div 60 = 20 \text{ minutes} \]

6. **Determine the next beep time**:
- The clocks beep together at 12:00 noon. After 20 minutes, they will beep together again at:
\[ 12:00 + 00:20 = 12:20 \text{ PM} \]

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

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