6 bells commence tolling together and toll of intervals are 2, 4, 6, 8, 10 and 12 s, respectively. In 1 h, how many times, do they toll together?

Options

A. 16
B. 32
C. 21
D. None of the above

Answer

To solve the problem of how many times the 6 bells toll together in 1 hour, we will follow these steps:

### Step 1: Identify the tolling intervals
The tolling intervals of the 6 bells are given as:
- Bell 1: 2 seconds
- Bell 2: 4 seconds
- Bell 3: 6 seconds
- Bell 4: 8 seconds
- Bell 5: 10 seconds
- Bell 6: 12 seconds

### Step 2: Find the Least Common Multiple (LCM)
To determine when all the bells toll together, we need to find the LCM of the intervals.

**Finding the LCM:**
- Prime factorization of each interval:
- 2 = 2
- 4 = 2^2
- 6 = 2 × 3
- 8 = 2^3
- 10 = 2 × 5
- 12 = 2^2 × 3

- The LCM is found by taking the highest power of each prime:
- For 2: max(1, 2, 1, 3, 1, 2) = 3 → 2^3
- For 3: max(0, 0, 1, 0, 0, 1) = 1 → 3^1
- For 5: max(0, 0, 0, 0, 1, 0) = 1 → 5^1

Thus, the LCM = 2^3 × 3^1 × 5^1 = 8 × 3 × 5 = 120 seconds.

### Step 3: Convert LCM to minutes
Since 120 seconds is equal to 2 minutes, this means the bells toll together every 2 minutes.

### Step 4: Calculate the number of times they toll together in 1 hour
1 hour = 60 minutes.

To find out how many times they toll together in 1 hour:
- Divide the total minutes in an hour by the interval at which they toll together:

Number of times = Total minutes / Tolling interval + 1 (for the initial toll)

Number of times = 60 minutes / 2 minutes + 1 = 30 + 1 = 31 times.

### Final Answer
The 6 bells toll together **31 times** in 1 hour.

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