An electronic divice makes a beep after every 15 minutes , Another device makes a beep after every 20 minutes. They beeped together at 6 a.m., at what time will they next beep together ?

Answer

To solve the problem of when the two devices will beep together next after 6 a.m., we need to find the least common multiple (LCM) of the two beeping intervals: 15 minutes and 20 minutes.

### Step-by-Step Solution:

1. **Identify the beeping intervals:**
- Device 1 beeps every 15 minutes.
- Device 2 beeps every 20 minutes.

2. **Find the LCM of 15 and 20:**
- To find the LCM, we can use the prime factorization method.
- The prime factorization of 15 is:
- \( 15 = 3 \times 5 \)
- The prime factorization of 20 is:
- \( 20 = 2^2 \times 5 \)

3. **List the highest powers of all prime factors:**
- From the factorizations, we have:
- For 2: the highest power is \( 2^2 \) (from 20)
- For 3: the highest power is \( 3^1 \) (from 15)
- For 5: the highest power is \( 5^1 \) (common in both)
- Therefore, the LCM is calculated as:
\[ LCM = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 \]

4. **Calculate the LCM:**
- First, calculate \( 4 \times 3 = 12 \).
- Then, calculate \( 12 \times 5 = 60 \).
- So, \( LCM(15, 20) = 60 \) minutes.

5. **Determine the next time they beep together:**
- Since they beeped together at 6 a.m., we add 60 minutes (1 hour) to this time.
- \( 6:00 \text{ a.m.} + 1 \text{ hour} = 7:00 \text{ a.m.} \)

### Final Answer:
The two devices will next beep together at **7:00 a.m.**.