Four bells toll at intervals of 5, 6, 8 and 12 seconds respectively. In how much time they will toll together.

Options

A. 120 sec.
B. 60 sec.
C. 90 sec.
D. 150 sec.

Answer

To find out when the four bells toll together, we need to determine the Least Common Multiple (LCM) of the intervals at which they toll: 5 seconds, 6 seconds, 8 seconds, and 12 seconds.

### Step 1: Prime Factorization
First, we will perform the prime factorization of each of the numbers.

- **5**: The prime factorization is \(5^1\).
- **6**: The prime factorization is \(2^1 \times 3^1\).
- **8**: The prime factorization is \(2^3\).
- **12**: The prime factorization is \(2^2 \times 3^1\).

### Step 2: Identify the Highest Powers of Each Prime
Next, we identify the highest power of each prime number that appears in the factorizations:

- For \(2\): The highest power is \(2^3\) (from 8).
- For \(3\): The highest power is \(3^1\) (from 6 and 12).
- For \(5\): The highest power is \(5^1\) (from 5).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying the highest powers of all prime factors together:

\[ \text{LCM} = 2^3 \times 3^1 \times 5^1 \]

Calculating this step-by-step:

1. Calculate \(2^3 = 8\).
2. Calculate \(3^1 = 3\).
3. Calculate \(5^1 = 5\).

Now multiply these results together:

\[ 8 \times 3 = 24 \]
\[ 24 \times 5 = 120 \]

Thus, the LCM is \(120\).

### Step 4: Conclusion
The four bells will toll together after \(120\) seconds.

### Final Answer
**The bells will toll together after 120 seconds.**

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