Four bells toll at intervals of 6,8, 12 and 18 minutes respectively If they start tolling together at 12 a.m Find after what interval will they toll together and how many times will they toll together in 6 hours ?

Options

A. 6 times
B. 5 times
C. 4 times
D. Data inadequate

Answer

To solve the problem, we need to find the least common multiple (LCM) of the intervals at which the bells toll: 6, 8, 12, and 18 minutes.

### Step 1: Find the prime factorization of each number.
- **6**: \( 2 \times 3 \)
- **8**: \( 2^3 \)
- **12**: \( 2^2 \times 3 \)
- **18**: \( 2 \times 3^2 \)

### Step 2: Identify the highest power of each prime factor.
- For the prime factor **2**: The highest power is \( 2^3 \) (from 8).
- For the prime factor **3**: The highest power is \( 3^2 \) (from 18).

### Step 3: Calculate the LCM using the highest powers.
\[ \text{LCM} = 2^3 \times 3^2 \]
Calculating this gives:
\[ = 8 \times 9 = 72 \]

### Step 4: Conclusion on the interval.
The bells will toll together every **72 minutes**.

### Step 5: Determine how many times they toll together in 6 hours.
6 hours is equal to \( 6 \times 60 = 360 \) minutes.

### Step 6: Calculate how many intervals of 72 minutes fit into 360 minutes.
\[ \text{Number of times} = \frac{360}{72} = 5 \]

### Final Answer:
The bells will toll together every **72 minutes**, and they will toll together **5 times** in 6 hours.

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