When Will Four Different Electronic Devices Beep Together Again?
In today's digital age, electronic devices often make sounds or 'beeps' to indicate various events or to alert users. Suppose you have four electronic devices that beep after every 30 minutes, 1 hour, 1 hour and 30 minutes, and 1 hour and 45 minutes, respectively. If all four devices beeped together at 12 noon, at what time will they beep together again?
Understanding the Problem
The intervals at which the four devices beep can be represented in minutes as follows:
30 minutes 90 minutes (1 hour and 30 minutes) 60 minutes (1 hour) 105 minutes (1 hour and 45 minutes)The key to solving this problem is to find the least common multiple (LCM) of these intervals. Finding the LCM will tell us the first time when all four devices will beep together again after the initial beep at 12 noon.
Solving the Problem: Finding the LCM
First, we need to factorize each of the given intervals:
30 2 x 3 x 5 90 2 x 3 x 3 x 5 60 2 x 2 x 3 x 5 105 3 x 5 x 7The LCM is found by taking the highest power of each prime factor that appears in the factorizations:
22 4 32 9 51 5 71 7So, the LCM of 30, 90, 60, and 105 is:
LCM 22 x 32 x 5 x 7 4 x 9 x 5 x 7 1260 minutes
Converting Minutes to Hours
Since 60 minutes make up an hour, we convert 1260 minutes to hours by dividing by 60:
1260 / 60 21 hours
Therefore, the four devices will beep together again 21 hours after 12 noon.
The Final Answer
The four electronic devices will beep together again at 9 am the next day.