Question

Four electronic devices make a beep after every 30 minutes, 1 hour, 1.5 hours and 1 hour 45 minutes respectively. All beeped together at 12 noon. At what time will they again beep together?

• A. 12 Midnight
• B. 3 AM
• C. 6 AM
• D. 9 AM

Hint:

Whenever multiple events repeat at fixed intervals, convert all intervals into the same unit and find their LCM to determine when they occur together again.

Answer 1

The Correct Option is D

Step 1: Convert all time intervals into minutes.
The first device beeps every 30 minutes.
The second device beeps every 1 hour, which is equal to 60 minutes.
The third device beeps every 1.5 hours, which is equal to 90 minutes.
The fourth device beeps every 1 hour 45 minutes, which is equal to 105 minutes.
Step 2: Write the intervals clearly.
The time intervals are:
30 minutes, 60 minutes, 90 minutes, and 105 minutes.
Step 3: Find the LCM of the given intervals.
Prime factorization:
\[ 30 = 2 \times 3 \times 5 \]
\[ 60 = 2^2 \times 3 \times 5 \]
\[ 90 = 2 \times 3^2 \times 5 \]
\[ 105 = 3 \times 5 \times 7 \]
Step 4: Take the highest powers of all prime factors.
\[ \text{LCM} = 2^2 \times 3^2 \times 5 \times 7 \]
\[ = 4 \times 9 \times 5 \times 7 \]
\[ = 1260 \text{ minutes} \]
Step 5: Convert LCM into hours.
\[ 1260 \div 60 = 21 \text{ hours} \]
Step 6: Add the time interval to the starting time.
All devices beeped together at 12 noon.
After 21 hours from 12 noon, the time will be:
\[ 12 \text{ noon} + 21 \text{ hours} = 9 \text{ AM (next day)} \]
Step 7: Final conclusion.
Hence, all four electronic devices will beep together again at 9 AM.