An electronic device makes a beep after every 60 sec. Another device makes a beep after every 62 sec. They beeped together at 10 a.m. The next time, when they would beep together at the earliest is

### Concept & Formula Simultaneous events occurring at different regular intervals repeat together at time durations that are common multiples of their individual intervals. The earliest next occurrence is found using the LCM: $$\\text{Time Interval} = \\text{LCM}(\\text{Interval}_1, \\text{Interval}_2)$$ ### Step-by-Step Solution **Given:** * Interval of first device = 60 seconds * Interval of second device = 62 seconds * Initial simultaneous beep time = 10:00 a.m. **Calculation:** 1. Find the LCM of 60 and 62: * $60 = 2^2 \\times 3 \\times 5$ * $62 = 2 \\times 31$ * $\\text{LCM} = 2^2 \\times 3 \\times 5 \\times 31 = 60 \\times 31 = 1860 \\text{ seconds}$ 2. Convert the total seconds into minutes: $$\\text{Minutes} = \\frac{1860}{60} = 31 \\text{ minutes}$$ 3. Add this interval to the initial starting time: $$\\text{Next Time} = 10:00 \\text{ a.m.} + 31 \\text{ minutes} = 10:31 \\text{ a.m.}$$ ### Exam Strategy & Shortcut Instead of computing the large multiplication fully, note that: $$\\text{LCM}(60, 62) = 60 \\times \\frac{62}{\\text{HCF}(60, 62)} = 60 \\times \\frac{62}{2} = 60 \\times 31 \\text{ seconds}$$ Dividing by 60 to get minutes immediately cancels out the 60, leaving exactly 31 minutes. Add 31 minutes directly to 10 a.m. to get 10:31 a.m. ### Common Pitfall A frequent error is forgetting to convert the calculated LCM from seconds to minutes, or making a manual error during the multiplication step ($60 \\times 31$). ### Final Answer **Therefore, the correct answer is 10.31 a.m.**