Six bells commence tolling together and toll at interval of 2, 4 , 6, 8 , 10 , 12 minutes respectively . In 30 hours , how many times do they toll together ?

Answer

To solve the problem of how many times six bells toll together in 30 hours, we can follow these steps:

### Step 1: Identify the tolling intervals
The bells toll at the following intervals:
- Bell 1: 2 minutes
- Bell 2: 4 minutes
- Bell 3: 6 minutes
- Bell 4: 8 minutes
- Bell 5: 10 minutes
- Bell 6: 12 minutes

### Step 2: Find the Least Common Multiple (LCM)
To determine how often all six bells toll together, we need to find the LCM of their tolling intervals (2, 4, 6, 8, 10, and 12).

- **Prime Factorization:**
- 2 = 2
- 4 = 2²
- 6 = 2 × 3
- 8 = 2³
- 10 = 2 × 5
- 12 = 2² × 3

- **Take the highest power of each prime:**
- For 2: the highest power is 2³ (from 8)
- For 3: the highest power is 3¹ (from 6 and 12)
- For 5: the highest power is 5¹ (from 10)

Thus, the LCM is:
\[ LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \]

### Step 3: Convert 30 hours into minutes
Since we need to calculate the number of times they toll together in minutes, we convert 30 hours into minutes:
\[ 30 \text{ hours} = 30 \times 60 = 1800 \text{ minutes} \]

### Step 4: Calculate how many times they toll together
To find how many times the bells toll together in 1800 minutes, we divide the total minutes by the LCM:
\[ \text{Number of times} = \frac{1800}{120} = 15 \]

### Step 5: Include the initial toll
Since the bells toll together at the start (time = 0), we need to add 1 to the count:
\[ \text{Total tolls together} = 15 + 1 = 16 \]

### Final Answer
Thus, the six bells toll together **16 times** in 30 hours.

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