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 at intervals of 20 and 25 minutes. They last beeped simultaneously at 12:00 PM. Determine the subsequent time they will beep together.
This requires calculating the Least Common Multiple (LCM) of 20 and 25 minutes.

Step 2: LCM Calculation:
The LCM is the smallest common multiple of two numbers.
Prime factorizations are:
\[20 = 2^2 \times 5\]\[25 = 5^2\]The LCM is computed using the highest power of each prime factor:
\[\text{LCM} = 2^2 \times 5^2 = 4 \times 25 = 100\]Therefore, the clocks will synchronize every 100 minutes.

Step 3: Time Advancement:
Adding the LCM (100 minutes) to the initial simultaneous beep time (12:00 PM) yields the next synchronization.
100 minutes is equivalent to 1 hour and 40 minutes.
Adding 1 hour and 40 minutes to 12:00 PM results in:
12:00 PM + 1 hour 40 minutes = 1:40 PM.

Step 4: Outcome:
The two alarm clocks will beep together again at 1:40 PM.