Four different electronic devices make a beep after every 30 minutes, 1 hour, $1\frac{1}{2}$ hour and 1 hour 45 minutes respectively. All the devices beeped together at 12 noon. They will again beep together at

### Concept & Formula To find when multiple periodic events will synchronize again, we calculate the Least Common Multiple (LCM) of their individual periods. $$\text{Next Simultaneous Event} = \text{Initial Time} + \text{LCM}(t_1, t_2, t_3, \dots)$$ Crucially, all time units must be converted to the same metric (e.g., minutes) before calculating the LCM. ### Step-by-Step Solution **Given:** * Device 1: 30 minutes * Device 2: 1 hour = 60 minutes * Device 3: $1\frac{1}{2}$ hour = 90 minutes * Device 4: 1 hour 45 minutes = 105 minutes * Start Time: 12 noon **Calculation:** 1. Find the LCM of 30, 60, 90, and 105: * $30 = 2 \times 3 \times 5$ * $60 = 2^2 \times 3 \times 5$ * $90 = 2 \times 3^2 \times 5$ * $105 = 3 \times 5 \times 7$ 2. Take the highest power of each prime factor: * $\text{LCM} = 2^2 \times 3^2 \times 5 \times 7 = 4 \times 9 \times 5 \times 7$ * $\text{LCM} = 1260 \text{ minutes}$ 3. Convert the LCM back into hours: $$\text{Hours} = \frac{1260}{60} = 21 \text{ hours}$$ 4. Add 21 hours to the starting time (12 noon): * 12 noon + 12 hours = 12 midnight * 12 midnight + 9 hours = 9 a.m. (the next day) ### Exam Strategy & Shortcut Instead of full prime factorization, use the step-by-step LCM method: * $\text{LCM}(30, 60) = 60$ * $\text{LCM}(60, 90) = 180$ * Now find $\text{LCM}(180, 105)$ * Both are divisible by 15: $180 = 15 \times 12$ and $105 = 15 \times 7$. * Since 12 and 7 are co-prime, $\text{LCM} = 15 \times 12 \times 7 = 180 \times 7 = 1260 \text{ minutes}$. Dividing 1260 by 60 mentally (126 / 6) gives 21 hours instantly. ### Common Pitfall Failing to correctly convert mixed time formats ($1\frac{1}{2}$ hours or 1 hour 45 minutes) strictly into minutes. Trying to find the LCM using decimal hours (like 1.5 or 1.75) without applying fraction LCM rules will lead to incorrect or messy calculations. ### Final Answer **Therefore, the correct answer is 9 a.m.**