Question

Two alarm clocks ring their alarms at regular intervals of 20 minutes and 25 minutes respectively. If they first beep together at 12 noon, at what time will they beep again together next time?

Answer 1

Step 1: Understanding the problem:

We are given that two alarm clocks beep at intervals of 20 minutes and 25 minutes. We need to find the next time they will beep together. This will happen at the Least Common Multiple (LCM) of 20 and 25 minutes.

Step 2: Finding the LCM of 20 and 25:

To find the LCM of 20 and 25, we can list the multiples of each number or use prime factorization.
The prime factorization of 20 is:
\[ 20 = 2^2 \times 5 \] The prime factorization of 25 is:
\[ 25 = 5^2 \] The LCM is found by taking the highest power of each prime factor that appears:
\[ \text{LCM} = 2^2 \times 5^2 = 4 \times 25 = 100 \] Thus, the LCM of 20 and 25 is \( 100 \) minutes.

Step 3: Calculating the next time they beep together:

Since the LCM is 100 minutes, the next time the clocks will beep together is 100 minutes after 12:00 PM.
100 minutes is equivalent to 1 hour and 40 minutes. Therefore, the clocks will beep together at 12:40 PM.

Step 4: Conclusion:

Thus, the next time the clocks beep together will be at 12:40 PM.