Four different electronic devices make a beep after every 30 minutes, 1 hour, `1(1)/(2)` hour and 1 hour 45 minutes respectively. All the devices beeped together at 12 noon.They will again beep together at:

Options

A. 12 midnght
B. 3 a.m.
C. 6 a.m.
D. 9 a.m.

Answer

To find out when the four electronic devices will beep together again after initially beeping at 12 noon, we need to determine the least common multiple (LCM) of the times each device takes to beep. Let's break down the solution step by step.

### Step 1: Convert the beep intervals to minutes
1. **Device 1:** Beeps every 30 minutes.
2. **Device 2:** Beeps every 1 hour = 60 minutes.
3. **Device 3:** Beeps every \(1 \frac{1}{2}\) hours = \(1 \times 60 + \frac{1}{2} \times 60 = 60 + 30 = 90\) minutes.
4. **Device 4:** Beeps every 1 hour 45 minutes = \(1 \times 60 + 45 = 60 + 45 = 105\) minutes.

### Step 2: List the beep intervals
- Device 1: 30 minutes
- Device 2: 60 minutes
- Device 3: 90 minutes
- Device 4: 105 minutes

### Step 3: Find the LCM of the beep intervals
To find the LCM, we first determine the prime factorization of each interval:
- **30:** \(2^1 \times 3^1 \times 5^1\)
- **60:** \(2^2 \times 3^1 \times 5^1\)
- **90:** \(2^1 \times 3^2 \times 5^1\)
- **105:** \(3^1 \times 5^1 \times 7^1\)

Now, we take the highest power of each prime factor:
- For \(2\): highest power is \(2^2\) (from 60)
- For \(3\): highest power is \(3^2\) (from 90)
- For \(5\): highest power is \(5^1\) (common in all)
- For \(7\): highest power is \(7^1\) (from 105)

### Step 4: Calculate the LCM
Now we calculate the LCM:
\[ \text{LCM} = 2^2 \times 3^2 \times 5^1 \times 7^1 = 4 \times 9 \times 5 \times 7 \]
Calculating step by step:
1. \(4 \times 9 = 36\)
2. \(36 \times 5 = 180\)
3. \(180 \times 7 = 1260\)

Thus, the LCM is **1260 minutes**.

### Step 5: Convert minutes back to hours
To convert 1260 minutes into hours:
\[ 1260 \div 60 = 21 \text{ hours} \]

### Step 6: Find the next beep time
Since the devices beeped together at 12 noon, we add 21 hours to this time:
- 12 noon + 21 hours = 9 AM the next day.

### Final Answer
The four devices will beep together again at **9 AM** the next day.

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