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 regular intervals of 20 minutes and 25 minutes respectively. They first beep together at 12 noon, and we need to find the next time they will beep together.
To solve this, we need to find the least common multiple (LCM) of the two time intervals: 20 minutes and 25 minutes.

Step 2: Finding the LCM of 20 and 25:
The LCM of two numbers is the smallest number that is a multiple of both.
The prime factorizations of 20 and 25 are:
\[ 20 = 2^2 \times 5 \] \[ 25 = 5^2 \] The LCM is found by taking the highest power of each prime factor:
\[ \text{LCM} = 2^2 \times 5^2 = 4 \times 25 = 100 \] So, the two clocks will beep together every 100 minutes.

Step 3: Adding the LCM to the start time:
Since they beep together at 12:00 noon, we add 100 minutes to this time to find when they will beep together next:
100 minutes is equal to 1 hour and 40 minutes.
So, adding 1 hour and 40 minutes to 12:00 noon gives:
12:00 + 1 hour 40 minutes = 1:40 PM.

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