An electronic device makes a beep after every 60 seconds. Another device makes a beep after every 62 seconds. They beeped together at 10 a.m . At what time will they beep together at the earliest ?

Explanation

To solve the problem of when the two electronic devices will beep together again after initially beeping together at 10 a.m., we need to find the Least Common Multiple (LCM) of the two beep intervals: 60 seconds and 62 seconds.

### Step-by-Step Solution:

1. **Identify the beep intervals**:
- Device 1 beeps every 60 seconds.
- Device 2 beeps every 62 seconds.

2. **Find the LCM of 60 and 62**:
- To find the LCM, we can use the prime factorization method.
- The prime factorization of 60 is:
\[ 60 = 2^2 \times 3^1 \times 5^1 \]
- The prime factorization of 62 is:
\[ 62 = 2^1 \times 31^1 \]
- Now, take the highest power of each prime number:
- For 2: \(2^2\)
- For 3: \(3^1\)
- For 5: \(5^1\)
- For 31: \(31^1\)

- Therefore, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^1 \times 31^1 = 4 \times 3 \times 5 \times 31 \]

3. **Calculate the LCM**:
- First, calculate \(4 \times 3 = 12\).
- Then, calculate \(12 \times 5 = 60\).
- Finally, calculate \(60 \times 31 = 1860\).
- Thus, \(LCM(60, 62) = 1860\) seconds.

4. **Convert LCM from seconds to minutes**:
- Since there are 60 seconds in a minute:
\[ \text{Minutes} = \frac{1860 \text{ seconds}}{60} = 31 \text{ minutes} \]

5. **Determine the next beep time**:
- The devices beeped together at 10:00 a.m.
- Adding 31 minutes to 10:00 a.m. gives:
\[ 10:00 \text{ a.m.} + 31 \text{ minutes} = 10:31 \text{ a.m.} \]

### Final Answer:
The two devices will beep together again at **10:31 a.m.**