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: Problem Definition:

Two alarm clocks beep every 20 minutes and 25 minutes, respectively. The objective is to determine the next instance they will beep simultaneously. This occurs at the Least Common Multiple (LCM) of their intervals.

Step 2: LCM Calculation:

The LCM of 20 and 25 is calculated using 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 derived by multiplying the highest power of each prime factor present in the factorizations: \[ \text{LCM} = 2^2 \times 5^2 = 4 \times 25 = 100 \] Therefore, the LCM of 20 and 25 minutes is 100 minutes.

Step 3: Determining the Next Simultaneous Beep:

With an LCM of 100 minutes, the clocks will beep together 100 minutes after 12:00 PM. 100 minutes is equivalent to 1 hour and 40 minutes. Consequently, the next simultaneous beep will occur at 12:40 PM.

Step 4: Conclusion:

The alarm clocks will next beep together at 12:40 PM.