Two alarm clocks ring their alarms at regular intervals of 72 seconds and 50 seconds if they first beep together at 12 noon, at what time will they beep again for the first time

Answer

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

### Step 1: Identify the intervals
The first alarm clock rings every 72 seconds, and the second alarm clock rings every 50 seconds.

### Step 2: Find the Least Common Multiple (LCM)
To determine when both clocks will beep together again, we need to find the LCM of 72 and 50.

- **Prime factorization of 72**:
- 72 = 2 × 36
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- So, the prime factorization of 72 is:
- \( 72 = 2^3 \times 3^2 \)

- **Prime factorization of 50**:
- 50 = 2 × 25
- 25 = 5 × 5
- So, the prime factorization of 50 is:
- \( 50 = 2^1 \times 5^2 \)

Now, we take the highest power of each prime factor:
- For \(2\): max(3, 1) = \(2^3\)
- For \(3\): max(2, 0) = \(3^2\)
- For \(5\): max(0, 2) = \(5^2\)

Thus, the LCM is:
\[ LCM(72, 50) = 2^3 \times 3^2 \times 5^2 \]

### Step 3: Calculate the LCM
Calculating the LCM:
\[ = 8 \times 9 \times 25 \]
\[ = 72 \times 25 = 1800 \text{ seconds} \]

### Step 4: Convert seconds to minutes
Now, convert 1800 seconds into minutes:
\[ 1800 \text{ seconds} = \frac{1800}{60} = 30 \text{ minutes} \]

### Step 5: Determine the time they beep together again
Since they first beeped together at 12 noon, we add 30 minutes to this time:
\[ 12:00 \text{ PM} + 30 \text{ minutes} = 12:30 \text{ PM} \]

### Conclusion
The two alarm clocks will beep together again for the first time at **12:30 PM**.

---