Four mobile phone commence vibrating together and vibrate at an intervals of 16 secs, 9 secs, 8 secs and 4 secs respectively . In 12 minutes, how many times will they vibrate together ?

Options

A. 7
B. 4
C. 8
D. None of these

Answer

To solve the problem step by step, let's follow the logical process:

### Step 1: Identify the intervals of vibration
The four mobile phones vibrate at intervals of:
- Phone 1: 16 seconds
- Phone 2: 9 seconds
- Phone 3: 8 seconds
- Phone 4: 4 seconds

### Step 2: Find the Least Common Multiple (LCM)
To determine how often all four phones vibrate together, we need to find the LCM of the intervals (16, 9, 8, and 4).

1. **Prime factorization**:
- 16 = 2^4
- 9 = 3^2
- 8 = 2^3
- 4 = 2^2

2. **Take the highest power of each prime**:
- For 2: the highest power is 2^4 (from 16)
- For 3: the highest power is 3^2 (from 9)

3. **Calculate the LCM**:
LCM = 2^4 × 3^2 = 16 × 9 = 144 seconds

### Step 3: Convert 12 minutes into seconds
Since we need to find out how many times they vibrate together in 12 minutes, we first convert the time into seconds.

12 minutes = 12 × 60 seconds = 720 seconds

### Step 4: Calculate how many times they vibrate together
Now, we divide the total time in seconds by the LCM to find out how many times they vibrate together.

Number of times they vibrate together = Total time in seconds / LCM
= 720 seconds / 144 seconds = 5

### Conclusion
Thus, the four mobile phones will vibrate together **5 times** in 12 minutes.

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